document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=408227>Question 408227</A>: <I><SPAN STYLE=\"word-break: break-all;\">  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #287678 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=287678\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> here are solutions for these problems...\r<BR>\n" );
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document.write( "a)\r<BR>\n" );
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document.write( "x^2-7x+12=0\r<BR>\n" );
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document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor x^2-7x+12=0, first multiply the leading coefficient 1 and the last term 12 to get 12. Now we need to ask ourselves: What two numbers multiply to 12 and add to -7? Lets find out by listing all of the possible factors of 12\r<BR>\n" );
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document.write( "Factors:\r<BR>\n" );
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document.write( "1,2,3,4,6,12,\r<BR>\n" );
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document.write( "-1,-2,-3,-4,-6,-12, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up to multiply to 12.\r<BR>\n" );
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document.write( "1*12=12\r<BR>\n" );
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document.write( "2*6=12\r<BR>\n" );
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document.write( "3*4=12\r<BR>\n" );
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document.write( "(-1)*(-12)=12\r<BR>\n" );
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document.write( "(-2)*(-6)=12\r<BR>\n" );
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document.write( "(-3)*(-4)=12\r<BR>\n" );
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document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
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document.write( "Now which of these pairs add to -7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -7\r<BR>\n" );
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document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	12	|	1+12=13<BR>\n" );
document.write( "2	|	6	|	2+6=8<BR>\n" );
document.write( "3	|	4	|	3+4=7<BR>\n" );
document.write( "-1	|	-12	|	-1+(-12)=-13<BR>\n" );
document.write( "-2	|	-6	|	-2+(-6)=-8<BR>\n" );
document.write( "-3	|	-4	|	-3+(-4)=-7\r<BR>\n" );
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document.write( "We can see from the table that -3 and -4 add to -7. So the two numbers that multiply to 12 and add to -7 are: -3 and -4\r<BR>\n" );
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document.write( "So the original quadratic\r<BR>\n" );
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document.write( "x^2-7x+12=0\r<BR>\n" );
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document.write( "breaks down to this (just replace -7x with the two numbers that multiply to 12 and add to -7, which are: -3 and -4)\r<BR>\n" );
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document.write( "x^2-3x-4x + 12 Replace -7x with -3x-4x\r<BR>\n" );
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document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
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document.write( "(x^2-3x) +(-4x + 12)\r<BR>\n" );
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document.write( "Factor a 1x out of the first group and factor a -4 out of the second group.\r<BR>\n" );
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document.write( "x(x-3) -4(-x - 3)\r<BR>\n" );
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document.write( "Now since we have a common term x-3 we can combine the two terms.\r<BR>\n" );
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document.write( "(x -4)(-x - 3) Combine like terms.<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the quadratic x^2-7x+12=0 factors to (x -4)(-x - 3) \r<BR>\n" );
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document.write( "Notice how  (x -4)(-x - 3)  foils back to our original problem x^2-7x+12=0. This verifies our answer. \r<BR>\n" );
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document.write( "b)<BR>\n" );
document.write( "x^2-10x+16=0<BR>\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor x^2-10x+16=0, first multiply the leading coefficient 1 and the last term 16 to get 16. Now we need to ask ourselves: What two numbers multiply to 16 and add to -10? Lets find out by listing all of the possible factors of 16\r<BR>\n" );
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document.write( "Factors:\r<BR>\n" );
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document.write( "1,2,4,8,16,\r<BR>\n" );
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document.write( "-1,-2,-4,-8,-16, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up to multiply to 16.\r<BR>\n" );
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document.write( "1*16=16\r<BR>\n" );
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document.write( "2*8=16\r<BR>\n" );
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document.write( "4*4=16\r<BR>\n" );
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document.write( "(-1)*(-16)=16\r<BR>\n" );
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document.write( "(-2)*(-8)=16\r<BR>\n" );
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document.write( "(-4)*(-4)=16\r<BR>\n" );
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document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
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document.write( "Now which of these pairs add to -10? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -10\r<BR>\n" );
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document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	16	|	1+16=17<BR>\n" );
document.write( "2	|	8	|	2+8=10<BR>\n" );
document.write( "4	|	4	|	4+4=8<BR>\n" );
document.write( "-1	|	-16	|	-1+(-16)=-17<BR>\n" );
document.write( "-2	|	-8	|	-2+(-8)=-10<BR>\n" );
document.write( "-4	|	-4	|	-4+(-4)=-8\r<BR>\n" );
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document.write( "We can see from the table that -2 and -8 add to -10. So the two numbers that multiply to 16 and add to -10 are: -2 and -8\r<BR>\n" );
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document.write( "So the original quadratic\r<BR>\n" );
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document.write( "\n" );
document.write( "x^2-10x+16=0\r<BR>\n" );
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document.write( "breaks down to this (just replace -10%2Ax with the two numbers that multiply to 16 and add to -10, which are: -2 and -8)\r<BR>\n" );
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document.write( "x^2-2x-8x+16=0 Replace -10x with -2x-8x\r<BR>\n" );
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document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
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document.write( "(x^2-2x)+(-8x+16)=0\r<BR>\n" );
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document.write( "Factor a 1x out of the first group and factor a -8 out of the second group.\r<BR>\n" );
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document.write( "x(x-2)-8(x-2)=0\r<BR>\n" );
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document.write( "Now since we have a common term x-2 we can combine the two terms.\r<BR>\n" );
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document.write( "(x-8)(x-2)=0 Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the quadratic x^2-2x-8x+16=0 factors to (x-8)(x-2)\r<BR>\n" );
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document.write( "Notice how (x-8)(x-2) foils back to our original problem x^2-2x-8x+16=0. This verifies our answer. \r<BR>\n" );
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document.write( "c)\r<BR>\n" );
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document.write( "x^2+2x-15=0\r<BR>\n" );
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document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor x^2+2x-15=0 , first multiply the leading coefficient 1 and the last term -15 to get -15. Now we need to ask ourselves: What two numbers multiply to -15 and add to 2? Lets find out by listing all of the possible factors of -15\r<BR>\n" );
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document.write( "Factors:\r<BR>\n" );
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document.write( "1,3,5,15,\r<BR>\n" );
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document.write( "-1,-3,-5,-15, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up to multiply to -15.\r<BR>\n" );
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document.write( "(-1)*(15)=-15\r<BR>\n" );
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document.write( "(-3)*(5)=-15\r<BR>\n" );
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document.write( "Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2\r<BR>\n" );
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document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	-15	|	1+(-15)=-14<BR>\n" );
document.write( "3	|	-5	|	3+(-5)=-2<BR>\n" );
document.write( "-1	|	15	|	(-1)+15=14<BR>\n" );
document.write( "-3	|	5	|	(-3)+5=2\r<BR>\n" );
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document.write( "We can see from the table that -3 and 5 add to 2. So the two numbers that multiply to -15 and add to 2 are: -3 and 5\r<BR>\n" );
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document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2+2x-15=0\r<BR>\n" );
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document.write( "breaks down to this (just replace 2x with the two numbers that multiply to -15 and add to 2, which are: -3 and 5)\r<BR>\n" );
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document.write( "\n" );
document.write( "x^2-3x+5x-15=0 Replace 2x with -3x+5x\r<BR>\n" );
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document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
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document.write( "(x^2-3x)+(5x-15)=0 \r<BR>\n" );
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document.write( "Factor a 1x out of the first group and factor a 5 out of the second group.\r<BR>\n" );
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document.write( "x(x-3)+5(x-3)=0 \r<BR>\n" );
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document.write( "Now since we have a common term x-3 we can combine the two terms.\r<BR>\n" );
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document.write( "(x+5)(x-3)=0  Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the quadratic x^2+2x-15=0 factors to (x+5)(x-3)\r<BR>\n" );
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document.write( "Notice how (x+5)(x-3) foils back to our original problem x^2+2x-15=0. This verifies our answer. \r<BR>\n" );
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document.write( "d)\r<BR>\n" );
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document.write( "x^2- 4x-21=0<BR>\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor x^2- 4x-21=0, first multiply the leading coefficient 1 and the last term -21 to get -21. Now we need to ask ourselves: What two numbers multiply to -21 and add to -4? Lets find out by listing all of the possible factors of -21\r<BR>\n" );
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document.write( "Factors:\r<BR>\n" );
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document.write( "1,3,7,21,\r<BR>\n" );
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document.write( "-1,-3,-7,-21, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up to multiply to -21.\r<BR>\n" );
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document.write( "(-1)*(21)=-21\r<BR>\n" );
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document.write( "(-3)*(7)=-21\r<BR>\n" );
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document.write( "Now which of these pairs add to -4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -4\r<BR>\n" );
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document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	-21	|	1+(-21)=-20<BR>\n" );
document.write( "3	|	-7	|	3+(-7)=-4<BR>\n" );
document.write( "-1	|	21	|	(-1)+21=20<BR>\n" );
document.write( "-3	|	7	|	(-3)+7=4\r<BR>\n" );
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document.write( "We can see from the table that 3 and -7 add to -4. So the two numbers that multiply to -21 and add to -4 are: 3 and -7\r<BR>\n" );
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document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2- 4x-21=0\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "breaks down to this (just replace -4x with the two numbers that multiply to -21 and add to -4, which are: 3 and -7)\r<BR>\n" );
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document.write( "\n" );
document.write( "x^2+3x-7x--21=0 .... Replace -4x with 3x-7x\r<BR>\n" );
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document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
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document.write( "(x^2+3x)+(-7x-21)\r<BR>\n" );
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document.write( "Factor a 1x out of the first group and factor a -7 out of the second group.\r<BR>\n" );
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document.write( "x(x-3)-7(x+3)\r<BR>\n" );
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document.write( "Now since we have a common term x+3 we can combine the two terms.\r<BR>\n" );
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document.write( "(x-7)(x+3)........ Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the quadratic x^2- 4x-21=0 factors to (x-7)(x+3)\r<BR>\n" );
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document.write( "<BR>\n" );
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document.write( "Notice how (x-7)(x+3) foils back to our original problem x^2- 4x-21=0. This verifies our answer. \r<BR>\n" );
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document.write( "e)\r<BR>\n" );
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document.write( "x^2-5x+6=0\r<BR>\n" );
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document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor x^2-5x+6=0, first multiply the leading coefficient 1 and the last term 6 to get 6. Now we need to ask ourselves: What two numbers multiply to 6 and add to -5? Lets find out by listing all of the possible factors of 6\r<BR>\n" );
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document.write( "Factors:\r<BR>\n" );
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document.write( "1,2,3,6,\r<BR>\n" );
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document.write( "-1,-2,-3,-6, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up to multiply to 6.\r<BR>\n" );
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document.write( "1*6=6\r<BR>\n" );
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document.write( "2*3=6\r<BR>\n" );
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document.write( "(-1)*(-6)=6\r<BR>\n" );
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document.write( "(-2)*(-3)=6\r<BR>\n" );
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document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
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document.write( "Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5\r<BR>\n" );
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document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	6	|	1+6=7<BR>\n" );
document.write( "2	|	3	|	2+3=5<BR>\n" );
document.write( "-1	|	-6	|	-1+(-6)=-7<BR>\n" );
document.write( "-2	|	-3	|	-2+(-3)=-5\r<BR>\n" );
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document.write( "We can see from the table that -2 and -3 add to -5. So the two numbers that multiply to 6 and add to -5 are: -2 and -3\r<BR>\n" );
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document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2-5x+6=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace -5x with the two numbers that multiply to 6 and add to -5, which are: -2 and -3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2-2x-3x+6=0 Replace -5x with -2x-3x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:<BR>\n" );
document.write( "(x^2-2x)+(-3x+6)=0\r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 1x out of the first group and factor a -3 out of the second group.\r<BR>\n" );
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document.write( "x(x-2)+3(-x+2)=0\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Now since we have a common term x-2 we can combine the two terms.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "(x+3)(-x+2)=0 Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the quadratic x^2-5x+6=0 factors (x+3)(-x+2)=0 \r<BR>\n" );
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document.write( "f)<BR>\n" );
document.write( "x^2+19x+18=0\r<BR>\n" );
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document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor x^2+19x+18=0, first multiply the leading coefficient 1 and the last term 18 to get 18. Now we need to ask ourselves: What two numbers multiply to 18 and add to 19? Lets find out by listing all of the possible factors of 18\r<BR>\n" );
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document.write( "Factors:\r<BR>\n" );
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document.write( "1,2,3,6,9,18,\r<BR>\n" );
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document.write( "-1,-2,-3,-6,-9,-18, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up to multiply to 18.\r<BR>\n" );
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document.write( "1*18=18\r<BR>\n" );
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document.write( "2*9=18\r<BR>\n" );
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document.write( "3*6=18\r<BR>\n" );
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document.write( "(-1)*(-18)=18\r<BR>\n" );
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document.write( "(-2)*(-9)=18\r<BR>\n" );
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document.write( "(-3)*(-6)=18\r<BR>\n" );
document.write( "\n" );
document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
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document.write( "Now which of these pairs add to 19? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 19\r<BR>\n" );
document.write( "\n" );
document.write( "||||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	18	|	1+18=19<BR>\n" );
document.write( "2	|	9	|	2+9=11<BR>\n" );
document.write( "3	|	6	|	3+6=9<BR>\n" );
document.write( "-1	|	-18	|	-1+(-18)=-19<BR>\n" );
document.write( "-2	|	-9	|	-2+(-9)=-11<BR>\n" );
document.write( "-3	|	-6	|	-3+(-6)=-9\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that 1 and 18 add to 19. So the two numbers that multiply to 18 and add to 19 are: 1 and 18\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2+19x+18=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace 19%2Ax with the two numbers that multiply to 18 and add to 19, which are: 1 and 18)\r<BR>\n" );
document.write( "\n" );
document.write( "x^2+1x +18x+18=0 Replace 19x with 1x+ 18x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "x^2+1x +18x+18=0\r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 1x out of the first group and factor a 18 out of the second group.\r<BR>\n" );
document.write( "\n" );
document.write( "x(x+1) +18(x+1)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term x+1 we can combine the two terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(x +18)(x+1)=0Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic x^2+19x+18=0 factors to (x +18)(x+1)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "g)<BR>\n" );
document.write( "x^2-17x+72=0\r<BR>\n" );
document.write( "\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor x^2-17x+72=0, first multiply the leading coefficient 1 and the last term 72 to get 72. Now we need to ask ourselves: What two numbers multiply to 72 and add to -17? Lets find out by listing all of the possible factors of 72\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,2,3,4,6,8,9,12,18,24,36,72,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to 72.\r<BR>\n" );
document.write( "\n" );
document.write( "1*72=72\r<BR>\n" );
document.write( "\n" );
document.write( "2*36=72\r<BR>\n" );
document.write( "\n" );
document.write( "3*24=72\r<BR>\n" );
document.write( "\n" );
document.write( "4*18=72\r<BR>\n" );
document.write( "\n" );
document.write( "6*12=72\r<BR>\n" );
document.write( "\n" );
document.write( "8*9=72\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(-72)=72\r<BR>\n" );
document.write( "\n" );
document.write( "(-2)*(-36)=72\r<BR>\n" );
document.write( "\n" );
document.write( "(-3)*(-24)=72\r<BR>\n" );
document.write( "\n" );
document.write( "(-4)*(-18)=72\r<BR>\n" );
document.write( "\n" );
document.write( "(-6)*(-12)=72\r<BR>\n" );
document.write( "\n" );
document.write( "(-8)*(-9)=72\r<BR>\n" );
document.write( "\n" );
document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to -17? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -17\r<BR>\n" );
document.write( "\n" );
document.write( "||||||||||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	72	|	1+72=73<BR>\n" );
document.write( "2	|	36	|	2+36=38<BR>\n" );
document.write( "3	|	24	|	3+24=27<BR>\n" );
document.write( "4	|	18	|	4+18=22<BR>\n" );
document.write( "6	|	12	|	6+12=18<BR>\n" );
document.write( "8	|	9	|	8+9=17<BR>\n" );
document.write( "-1	|	-72	|	-1+(-72)=-73<BR>\n" );
document.write( "-2	|	-36	|	-2+(-36)=-38<BR>\n" );
document.write( "-3	|	-24	|	-3+(-24)=-27<BR>\n" );
document.write( "-4	|	-18	|	-4+(-18)=-22<BR>\n" );
document.write( "-6	|	-12	|	-6+(-12)=-18<BR>\n" );
document.write( "-8	|	-9	|	-8+(-9)=-17\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that -8 and -9 add to -17. So the two numbers that multiply to 72 and add to -17 are: -8 and -9\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2-17x+72=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace -17x with the two numbers that multiply to 72 and add to -17, which are: -8 and -9)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2-8x - 9x+72=0 Replace -17x with -8x-9x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "(x^2-8x)+( - 9x+72)=0 \r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 1x out of the first group and factor a -9 out of the second group.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x(x-8)-9( x-8)=0 \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term x-8 we can combine the two terms.\r<BR>\n" );
document.write( "\n" );
document.write( "(x-9)( x-8)=0  Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic x^2-17x+72=0 factors to (x-9)( x-8)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "h)\r<BR>\n" );
document.write( "\n" );
document.write( "x^2+5x=0\r<BR>\n" );
document.write( "\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "Notice how each term in the expression x^2+5x=0\r<BR>\n" );
document.write( "\n" );
document.write( "has a common factor of x. We can simply factor out an x like this:<BR>\n" );
document.write( "x(x+5)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i)\r<BR>\n" );
document.write( "\n" );
document.write( "x^2+8x+7=0\r<BR>\n" );
document.write( "\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor x^2+8x+7=0, first multiply the leading coefficient 1 and the last term 7 to get 7. Now we need to ask ourselves: What two numbers multiply to 7 and add to 8? Lets find out by listing all of the possible factors of 7\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,7,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-7, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to 7.\r<BR>\n" );
document.write( "\n" );
document.write( "1*7=7\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(-7)=7\r<BR>\n" );
document.write( "\n" );
document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to 8? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 8\r<BR>\n" );
document.write( "\n" );
document.write( "||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	7	|	1+7=8<BR>\n" );
document.write( "-1	|	-7	|	-1+(-7)=-8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that 1 and 7 add to 8. So the two numbers that multiply to 7 and add to 8 are: 1 and 7\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2+8x+7=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace 8x with the two numbers that multiply to 7 and add to 8, which are: 1 and 7)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2+x+7x+7=0 Replace 8x with x+7x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "(x^2+x)+(7x+7)=0 \r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 1x out of the first group and factor a 7 out of the second group.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x(x+1)+7(x+1)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term x+1 we can combine the two terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(x+7)(x+1)=0Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic x^2+8x+7=0 factors to (x+7)(x+1)=0\r<BR>\n" );
document.write( "\n" );
document.write( "j)\r<BR>\n" );
document.write( "\n" );
document.write( "3x^2+6x=0\r<BR>\n" );
document.write( "\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "Notice how each term in the expression 3x^2+6x=0\r<BR>\n" );
document.write( "\n" );
document.write( "has a common factor of 3x. We can simply factor out an 3x like this:<BR>\n" );
document.write( "3x(x+2)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "k)\r<BR>\n" );
document.write( "\n" );
document.write( "2x^2=32-12x.....move all terms to the left\r<BR>\n" );
document.write( "\n" );
document.write( "2x^2+12x-32=0.........divide each term by 2\r<BR>\n" );
document.write( "\n" );
document.write( "x^2+6x-16=0\r<BR>\n" );
document.write( "\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor x^2+6x-16=0, first multiply the leading coefficient 1 and the last term 16 to get 16. Now we need to ask ourselves: What two numbers multiply to 16 and add to 6? Lets find out by listing all of the possible factors of 16\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,2,4,8,16,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-2,-4,-8,-16, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to 16.\r<BR>\n" );
document.write( "\n" );
document.write( "1*16=16\r<BR>\n" );
document.write( "\n" );
document.write( "2*8=16\r<BR>\n" );
document.write( "\n" );
document.write( "4*4=16\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(-16)=16\r<BR>\n" );
document.write( "\n" );
document.write( "(-2)*(-8)=16\r<BR>\n" );
document.write( "\n" );
document.write( "(-4)*(-4)=16\r<BR>\n" );
document.write( "\n" );
document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to 6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 6\r<BR>\n" );
document.write( "\n" );
document.write( "||||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	16	|	1+16=17<BR>\n" );
document.write( "2	|	8	|	2+8=10<BR>\n" );
document.write( "4	|	4	|	4+4=8<BR>\n" );
document.write( "-1	|	-16	|	-1+(-16)=-17<BR>\n" );
document.write( "-2	|	-8	|	-2+(-8)=-10<BR>\n" );
document.write( "-4	|	-4	|	-4+(-4)=-8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "None of these factors add to 6. So the quadratic x^2+6x-16=0 cannot be factored. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "l)\r<BR>\n" );
document.write( "\n" );
document.write( "3x^2+7x-6=0<BR>\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor 3x^2+7x-6=0, first multiply the leading coefficient 3 and the last term -6 to get -18. Now we need to ask ourselves: What two numbers multiply to -18 and add to 7? Lets find out by listing all of the possible factors of -18\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,2,3,6,9,18,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-2,-3,-6,-9,-18, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to -18.\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(18)=-18\r<BR>\n" );
document.write( "\n" );
document.write( "(-2)*(9)=-18\r<BR>\n" );
document.write( "\n" );
document.write( "(-3)*(6)=-18\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7\r<BR>\n" );
document.write( "\n" );
document.write( "||||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	-18	|	1+(-18)=-17<BR>\n" );
document.write( "2	|	-9	|	2+(-9)=-7<BR>\n" );
document.write( "3	|	-6	|	3+(-6)=-3<BR>\n" );
document.write( "-1	|	18	|	(-1)+18=17<BR>\n" );
document.write( "-2	|	9	|	(-2)+9=7<BR>\n" );
document.write( "-3	|	6	|	(-3)+6=3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that -2 and 9 add to 7. So the two numbers that multiply to -18 and add to 7 are: -2 and 9\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3x^2+7x-6=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace 7x with the two numbers that multiply to -18 and add to 7, which are: -2 and 9)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3x^2-2x+9x-6=0... Replace 7x with -2x+9x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "(3x^2-2x)+(9x-6)=0\r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 1x out of the first group and factor a 3 out of the second group.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x(3x-2)+3(3x-2)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term 3x-2 we can combine the two terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(x+3)(3x-2)=0.......... Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic 3x^2+7x-6=0 factors to (x+3)(3x-2)=0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "m)\r<BR>\n" );
document.write( "\n" );
document.write( "x^2=10x-21............move all terms to the left\r<BR>\n" );
document.write( "\n" );
document.write( "x^2-10x+21=0\r<BR>\n" );
document.write( "\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor x^2-10x+21=0, first multiply the leading coefficient 1 and the last term 21 to get 21. Now we need to ask ourselves: What two numbers multiply to 21 and add to -10? Lets find out by listing all of the possible factors of 21\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,3,7,21,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-3,-7,-21, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to 21.\r<BR>\n" );
document.write( "\n" );
document.write( "1*21=21\r<BR>\n" );
document.write( "\n" );
document.write( "3*7=21\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(-21)=21\r<BR>\n" );
document.write( "\n" );
document.write( "(-3)*(-7)=21\r<BR>\n" );
document.write( "\n" );
document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to -10? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -10\r<BR>\n" );
document.write( "\n" );
document.write( "||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	21	|	1+21=22<BR>\n" );
document.write( "3	|	7	|	3+7=10<BR>\n" );
document.write( "-1	|	-21	|	-1+(-21)=-22<BR>\n" );
document.write( "-3	|	-7	|	-3+(-7)=-10\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that -3 and -7 add to -10. So the two numbers that multiply to 21 and add to -10 are: -3 and -7\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2-10x+21=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace -10%2Ax with the two numbers that multiply to 21 and add to -10, which are: -3 and -7)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2-3x-7x+21=0 Replace -10x with -3x-7x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "(x^2-3x)+(-7x+21)=0\r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 1x out of the first group and factor a -7 out of the second group.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x(x-3)-7(x-3)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term x-3 we can combine the two terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(x-7)(x-3)=0.......Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic x^2-10x+21=0 factors to (x-7)(x-3)=0\r<BR>\n" );
document.write( "\n" );
document.write( "n)\r<BR>\n" );
document.write( "\n" );
document.write( "x^2-x-56=0\r<BR>\n" );
document.write( "\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor x^2-x-56=0, first multiply the leading coefficient 1 and the last term -56 to get -56. Now we need to ask ourselves: What two numbers multiply to -56 and add to -1? Lets find out by listing all of the possible factors of -56\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,2,4,7,8,14,28,56,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-2,-4,-7,-8,-14,-28,-56, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to -56.\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(56)=-56\r<BR>\n" );
document.write( "\n" );
document.write( "(-2)*(28)=-56\r<BR>\n" );
document.write( "\n" );
document.write( "(-4)*(14)=-56\r<BR>\n" );
document.write( "\n" );
document.write( "(-7)*(8)=-56\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1\r<BR>\n" );
document.write( "\n" );
document.write( "||||||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	-56	|	1+(-56)=-55<BR>\n" );
document.write( "2	|	-28	|	2+(-28)=-26<BR>\n" );
document.write( "4	|	-14	|	4+(-14)=-10<BR>\n" );
document.write( "7	|	-8	|	7+(-8)=-1<BR>\n" );
document.write( "-1	|	56	|	(-1)+56=55<BR>\n" );
document.write( "-2	|	28	|	(-2)+28=26<BR>\n" );
document.write( "-4	|	14	|	(-4)+14=10<BR>\n" );
document.write( "-7	|	8	|	(-7)+8=1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that 7 and -8 add to -1. So the two numbers that multiply to -56 and add to -1 are: 7 and -8\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2-x-56=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace -1x with the two numbers that multiply to -56 and add to -1, which are: 7 and -8)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2+7x-8x-56=0 Replace -1x with 7x-8x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "(x^2+7x)+(-8x-56)=0\r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 1x out of the first group and factor a -8 out of the second group.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x(x+7)-8(x+7)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term x+7 we can combine the two terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(x-8)(x+7)=0..... Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic x^2-x-56=0 factors to (x-8)(x+7)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "o)\r<BR>\n" );
document.write( "\n" );
document.write( "3x^2+2x-5=0\r<BR>\n" );
document.write( "\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor 3x^2+2x-5=0, first multiply the leading coefficient 3 and the last term -5 to get -15. Now we need to ask ourselves: What two numbers multiply to -15 and add to 2? Lets find out by listing all of the possible factors of -15\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,3,5,15,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-3,-5,-15, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to -15.\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(15)=-15\r<BR>\n" );
document.write( "\n" );
document.write( "(-3)*(5)=-15\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2\r<BR>\n" );
document.write( "\n" );
document.write( "||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	-15	|	1+(-15)=-14<BR>\n" );
document.write( "3	|	-5	|	3+(-5)=-2<BR>\n" );
document.write( "-1	|	15	|	(-1)+15=14<BR>\n" );
document.write( "-3	|	5	|	(-3)+5=2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that -3 and 5 add to 2. So the two numbers that multiply to -15 and add to 2 are: -3 and 5\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "\n" );
document.write( "3x^2+2x-5=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace 2x with the two numbers that multiply to -15 and add to 2, which are: -3 and 5)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3x^2-3x+5x-5=0.... Replace 2x with -3x+5x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:<BR>\n" );
document.write( "(3x^2-3x)+(5x-5)=0.\r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 3x out of the first group and factor a 5 out of the second group.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3x(x-1)+5(x-1)=0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term x-1 we can combine the two terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(3x+5)(x-1)=0. ..Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic 3x^2+2x-5=0 factors to (3x+5)(x-1)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "p)\r<BR>\n" );
document.write( "\n" );
document.write( "6x^2=6-5x...move all terms to the left\r<BR>\n" );
document.write( "\n" );
document.write( "6x^2+5x-6=0<BR>\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor 6x^2+5x-6=0, first multiply the leading coefficient 6 and the last term -6 to get -36. Now we need to ask ourselves: What two numbers multiply to -36 and add to 5? Lets find out by listing all of the possible factors of -36\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,2,3,4,6,9,12,18,36,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-36, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to -36.\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(36)=-36\r<BR>\n" );
document.write( "\n" );
document.write( "(-2)*(18)=-36\r<BR>\n" );
document.write( "\n" );
document.write( "(-3)*(12)=-36\r<BR>\n" );
document.write( "\n" );
document.write( "(-4)*(9)=-36\r<BR>\n" );
document.write( "\n" );
document.write( "(-6)*(6)=-36\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5\r<BR>\n" );
document.write( "\n" );
document.write( "||||||||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	-36	|	1+(-36)=-35<BR>\n" );
document.write( "2	|	-18	|	2+(-18)=-16<BR>\n" );
document.write( "3	|	-12	|	3+(-12)=-9<BR>\n" );
document.write( "4	|	-9	|	4+(-9)=-5<BR>\n" );
document.write( "6	|	-6	|	6+(-6)=0<BR>\n" );
document.write( "-1	|	36	|	(-1)+36=35<BR>\n" );
document.write( "-2	|	18	|	(-2)+18=16<BR>\n" );
document.write( "-3	|	12	|	(-3)+12=9<BR>\n" );
document.write( "-4	|	9	|	(-4)+9=5<BR>\n" );
document.write( "-6	|	6	|	(-6)+6=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that -4 and 9 add to 5. So the two numbers that multiply to -36 and add to 5 are: -4 and 9\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "\n" );
document.write( "6x^2+5x-6=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace 5%2Ax with the two numbers that multiply to -36 and add to 5, which are: -4 and 9)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "6x^2-4x+9x-6=0 Replace 5x with -4x+9x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "(6x^2-4x)+(9x-6)=0\r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 2x out of the first group and factor a 3 out of the second group.\r<BR>\n" );
document.write( "\n" );
document.write( "2x(3x-2)+3(3x-2)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term 3x-2 we can combine the two terms.\r<BR>\n" );
document.write( "\n" );
document.write( "(2x+3)(3x-2)=0..........Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic 6x^2+5x-6=0 factors to (2x+3)(3x-2)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "q)\r<BR>\n" );
document.write( "\n" );
document.write( "4x^2+19x=5<BR>\n" );
document.write( "4x^2+19x-5=0\r<BR>\n" );
document.write( "\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor 4x^2+19x-5=0, first multiply the leading coefficient 4 and the last term -5 to get -20. Now we need to ask ourselves: What two numbers multiply to -20 and add to 19? Lets find out by listing all of the possible factors of -20\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,2,4,5,10,20,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-2,-4,-5,-10,-20, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to -20.\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(20)=-20\r<BR>\n" );
document.write( "\n" );
document.write( "(-2)*(10)=-20\r<BR>\n" );
document.write( "\n" );
document.write( "(-4)*(5)=-20\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to 19? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 19\r<BR>\n" );
document.write( "\n" );
document.write( "||||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	-20	|	1+(-20)=-19<BR>\n" );
document.write( "2	|	-10	|	2+(-10)=-8<BR>\n" );
document.write( "4	|	-5	|	4+(-5)=-1<BR>\n" );
document.write( "-1	|	20	|	(-1)+20=19<BR>\n" );
document.write( "-2	|	10	|	(-2)+10=8<BR>\n" );
document.write( "-4	|	5	|	(-4)+5=1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that -1 and 20 add to 19. So the two numbers that multiply to -20 and add to 19 are: -1 and 20\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4x^2+19x-5=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace 19x with the two numbers that multiply to -20 and add to 19, which are: -1 and 20)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4x^2-x+20x-5=0 ..........Replace 19x with -x+20x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "(4x^2-x)+(20x-5)=0 \r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 1x out of the first group and factor a 5 out of the second group.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x(4x-1)+5(4x-1)=0 \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term 4x-1 we can combine the two terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(x+5)(4x-1)=0 .......... Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic 4x^2+19x-5=0 factors to (x+5)(4x-1)=0 \r<BR>\n" );
document.write( "\n" );
document.write( "r)\r<BR>\n" );
document.write( "\n" );
document.write( "6x^2-24=0\r<BR>\n" );
document.write( "\n" );
document.write( "6(x^2-4)=0............6 is not equal to zero, so (x^2-4)=0\r<BR>\n" );
document.write( "\n" );
document.write( "6(x^2-2^2)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "6(x-2)(x+2)=0\r<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic 6x^2-24=0 factors to 6(x-2)(x+2)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "s)\r<BR>\n" );
document.write( "\n" );
document.write( "4x^2-9=0\r<BR>\n" );
document.write( "\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor 4x^2-9=0, first multiply the leading coefficient 4 and the last term -9 to get -36. Now we need to ask ourselves: What two numbers multiply to -36 and add to 0? Lets find out by listing all of the possible factors of -36\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,2,3,4,6,9,12,18,36,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-36, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to -36.\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(36)=-36\r<BR>\n" );
document.write( "\n" );
document.write( "(-2)*(18)=-36\r<BR>\n" );
document.write( "\n" );
document.write( "(-3)*(12)=-36\r<BR>\n" );
document.write( "\n" );
document.write( "(-4)*(9)=-36\r<BR>\n" );
document.write( "\n" );
document.write( "(-6)*(6)=-36\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to 0? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 0\r<BR>\n" );
document.write( "\n" );
document.write( "||||||||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	-36	|	1+(-36)=-35<BR>\n" );
document.write( "2	|	-18	|	2+(-18)=-16<BR>\n" );
document.write( "3	|	-12	|	3+(-12)=-9<BR>\n" );
document.write( "4	|	-9	|	4+(-9)=-5<BR>\n" );
document.write( "6	|	-6	|	6+(-6)=0<BR>\n" );
document.write( "-1	|	36	|	(-1)+36=35<BR>\n" );
document.write( "-2	|	18	|	(-2)+18=16<BR>\n" );
document.write( "-3	|	12	|	(-3)+12=9<BR>\n" );
document.write( "-4	|	9	|	(-4)+9=5<BR>\n" );
document.write( "-6	|	6	|	(-6)+6=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that 6 and -6 add to 0. So the two numbers that multiply to -36 and add to 0 are: 6 and -6\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4x^2-9=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace 0x with the two numbers that multiply to -36 and add to 0, which are: 6 and -6)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4x^2+ 6x-6x -9=0 ............Replace 0x with 6x-6x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "(4x^2+ 6x)+(-6x -9)=0 \r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 2x out of the first group and factor a -3 out of the second group.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2x(2x+ 3)-3(2x +3)=0 \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term 2x+3 we can combine the two terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(2x-3)(2x +3)=0 ... Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic 4x^2-9=0 factors to (2x-3)(2x +3)=0 \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "t)\r<BR>\n" );
document.write( "\n" );
document.write( "2x^2=13x-20\r<BR>\n" );
document.write( "\n" );
document.write( "2x^2-13x+20=0\r<BR>\n" );
document.write( "\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor 2x^2-13x+20=0, first multiply the leading coefficient 2 and the last term 20 to get 40. Now we need to ask ourselves: What two numbers multiply to 40 and add to -13? Lets find out by listing all of the possible factors of 40\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,2,4,5,8,10,20,40,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-2,-4,-5,-8,-10,-20,-40, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to 40.\r<BR>\n" );
document.write( "\n" );
document.write( "1*40=40\r<BR>\n" );
document.write( "\n" );
document.write( "2*20=40\r<BR>\n" );
document.write( "\n" );
document.write( "4*10=40\r<BR>\n" );
document.write( "\n" );
document.write( "5*8=40\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(-40)=40\r<BR>\n" );
document.write( "\n" );
document.write( "(-2)*(-20)=40\r<BR>\n" );
document.write( "\n" );
document.write( "(-4)*(-10)=40\r<BR>\n" );
document.write( "\n" );
document.write( "(-5)*(-8)=40\r<BR>\n" );
document.write( "\n" );
document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to -13? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -13\r<BR>\n" );
document.write( "\n" );
document.write( "||||||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	40	|	1+40=41<BR>\n" );
document.write( "2	|	20	|	2+20=22<BR>\n" );
document.write( "4	|	10	|	4+10=14<BR>\n" );
document.write( "5	|	8	|	5+8=13<BR>\n" );
document.write( "-1	|	-40	|	-1+(-40)=-41<BR>\n" );
document.write( "-2	|	-20	|	-2+(-20)=-22<BR>\n" );
document.write( "-4	|	-10	|	-4+(-10)=-14<BR>\n" );
document.write( "-5	|	-8	|	-5+(-8)=-13\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that -5 and -8 add to -13. So the two numbers that multiply to 40 and add to -13 are: -5 and -8\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2x^2-13x+20=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace -13x with the two numbers that multiply to 40 and add to -13, which are: -5 and -8)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2x^2-5x-8x+20=0........... Replace -13x with -5x-8x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "(2x^2-5x)+(-8x+20)=0\r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 1x out of the first group and factor a -4 out of the second group.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x(2x-5)-4(2x+5)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term 2x-5 we can combine the two terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(x-4)(2x+5)=0.... Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic 2x^2-13x+20=0 factors to (x-4)(2x+5)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "u)\r<BR>\n" );
document.write( "\n" );
document.write( "12x^2+7x=12<BR>\n" );
document.write( "12x^2+7x-12=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Solution by Factoring using the AC method (Factor by Grouping)<BR>\n" );
document.write( "In order to factor 12x^2+7x-12=0, first multiply the leading coefficient 12 and the last term -12 to get -144. Now we need to ask ourselves: What two numbers multiply to -144 and add to 7? Lets find out by listing all of the possible factors of -144\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,2,3,4,6,8,9,12,16,18,24,36,48,72,144,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72,-144, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to -144.\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(144)=-144\r<BR>\n" );
document.write( "\n" );
document.write( "(-2)*(72)=-144\r<BR>\n" );
document.write( "\n" );
document.write( "(-3)*(48)=-144\r<BR>\n" );
document.write( "\n" );
document.write( "(-4)*(36)=-144\r<BR>\n" );
document.write( "\n" );
document.write( "(-6)*(24)=-144\r<BR>\n" );
document.write( "\n" );
document.write( "(-8)*(18)=-144\r<BR>\n" );
document.write( "\n" );
document.write( "(-9)*(16)=-144\r<BR>\n" );
document.write( "\n" );
document.write( "(-12)*(12)=-144\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7\r<BR>\n" );
document.write( "\n" );
document.write( "||||||||||||||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	-144	|	1+(-144)=-143<BR>\n" );
document.write( "2	|	-72	|	2+(-72)=-70<BR>\n" );
document.write( "3	|	-48	|	3+(-48)=-45<BR>\n" );
document.write( "4	|	-36	|	4+(-36)=-32<BR>\n" );
document.write( "6	|	-24	|	6+(-24)=-18<BR>\n" );
document.write( "8	|	-18	|	8+(-18)=-10<BR>\n" );
document.write( "9	|	-16	|	9+(-16)=-7<BR>\n" );
document.write( "12	|	-12	|	12+(-12)=0<BR>\n" );
document.write( "-1	|	144	|	(-1)+144=143<BR>\n" );
document.write( "-2	|	72	|	(-2)+72=70<BR>\n" );
document.write( "-3	|	48	|	(-3)+48=45<BR>\n" );
document.write( "-4	|	36	|	(-4)+36=32<BR>\n" );
document.write( "-6	|	24	|	(-6)+24=18<BR>\n" );
document.write( "-8	|	18	|	(-8)+18=10<BR>\n" );
document.write( "-9	|	16	|	(-9)+16=7<BR>\n" );
document.write( "-12	|	12	|	(-12)+12=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that -9 and 16 add to 7. So the two numbers that multiply to -144 and add to 7 are: -9 and 16\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "\n" );
document.write( "12x^2+7x-12=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace 7%2Ax with the two numbers that multiply to -144 and add to 7, which are: -9 and 16)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "12x^2-9x+16x-12=0... Replace 7x with -9x+16x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "(12x^2-9x)+(16x-12)=0\r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 3x out of the first group and factor a 4 out of the second group.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3x(4x-3)+4(4x-3)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term 4x-3 we can combine the two terms.\r<BR>\n" );
document.write( "\n" );
document.write( "(3x+4)(4x-3)=0........ Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic 12x^2+7x-12=0 factors to (3x+4)(4x-3)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "v)\r<BR>\n" );
document.write( "\n" );
document.write( "6x^2+29x-5=0\r<BR>\n" );
document.write( "\n" );
document.write( "In order to factor 6x^2+29x-5=0, first multiply the leading coefficient 6 and the last term -5 to get -30. Now we need to ask ourselves: What two numbers multiply to -30 and add to 29? Lets find out by listing all of the possible factors of -30\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,2,3,5,6,10,15,30,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-2,-3,-5,-6,-10,-15,-30, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to -30.\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(30)=-30\r<BR>\n" );
document.write( "\n" );
document.write( "(-2)*(15)=-30\r<BR>\n" );
document.write( "\n" );
document.write( "(-3)*(10)=-30\r<BR>\n" );
document.write( "\n" );
document.write( "(-5)*(6)=-30\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to 29? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 29\r<BR>\n" );
document.write( "\n" );
document.write( "||||||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	-30	|	1+(-30)=-29<BR>\n" );
document.write( "2	|	-15	|	2+(-15)=-13<BR>\n" );
document.write( "3	|	-10	|	3+(-10)=-7<BR>\n" );
document.write( "5	|	-6	|	5+(-6)=-1<BR>\n" );
document.write( "-1	|	30	|	(-1)+30=29<BR>\n" );
document.write( "-2	|	15	|	(-2)+15=13<BR>\n" );
document.write( "-3	|	10	|	(-3)+10=7<BR>\n" );
document.write( "-5	|	6	|	(-5)+6=1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that -1 and 30 add to 29. So the two numbers that multiply to -30 and add to 29 are: -1 and 30\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "6x^2+29x-5=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace 29x with the two numbers that multiply to -30 and add to 29, which are: -1 and 30)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "6x^2-x+30x-5=0..... Replace 29x with -x+30x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "(6x^2-x)+(30x-5)=0\r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 1x out of the first group and factor a 5 out of the second group.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x(6x-1)+5(6x-1)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term 6x-1 we can combine the two terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(x+5)(6x-1)=0.........Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic 6x^2+29x-5=0 factors to (x+5)(6x-1)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "w)\r<BR>\n" );
document.write( "\n" );
document.write( "(x-5)^2=36..........take a square root\r<BR>\n" );
document.write( "\n" );
document.write( "sqrt((x-5)^2)=sqrt(36)\r<BR>\n" );
document.write( "\n" );
document.write( "x-5=6\r<BR>\n" );
document.write( "\n" );
document.write( "x-5-6=0\r<BR>\n" );
document.write( "\n" );
document.write( "x-11=0.........\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x)\r<BR>\n" );
document.write( "\n" );
document.write( "(x+2)(x+3)=2\r<BR>\n" );
document.write( "\n" );
document.write( "(x+2)(x+3)-2=0\r<BR>\n" );
document.write( "\n" );
document.write( "x^2+3x+2x+6-2=0\r<BR>\n" );
document.write( "\n" );
document.write( "x^2+5x+4=0\r<BR>\n" );
document.write( "\n" );
document.write( "In order to factor x^2+5x+4=0, first multiply the leading coefficient 1 and the last term 4 to get 4. Now we need to ask ourselves: What two numbers multiply to 4 and add to 5? Lets find out by listing all of the possible factors of 4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1,2,4,\r<BR>\n" );
document.write( "\n" );
document.write( "-1,-2,-4, List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
document.write( "\n" );
document.write( "These factors pair up to multiply to 4.\r<BR>\n" );
document.write( "\n" );
document.write( "1*4=4\r<BR>\n" );
document.write( "\n" );
document.write( "2*2=4\r<BR>\n" );
document.write( "\n" );
document.write( "(-1)*(-4)=4\r<BR>\n" );
document.write( "\n" );
document.write( "(-2)*(-2)=4\r<BR>\n" );
document.write( "\n" );
document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
document.write( "\n" );
document.write( "Now which of these pairs add to 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5\r<BR>\n" );
document.write( "\n" );
document.write( "||||<BR>\n" );
document.write( "First Number	|	Second Number	|	Sum<BR>\n" );
document.write( "1	|	4	|	1+4=5<BR>\n" );
document.write( "2	|	2	|	2+2=4<BR>\n" );
document.write( "-1	|	-4	|	-1+(-4)=-5<BR>\n" );
document.write( "-2	|	-2	|	-2+(-2)=-4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can see from the table that 1 and 4 add to 5. So the two numbers that multiply to 4 and add to 5 are: 1 and 4\r<BR>\n" );
document.write( "\n" );
document.write( "So the original quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x^2+5x+4=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "breaks down to this (just replace 5x with the two numbers that multiply to 4 and add to 5, which are: 1 and 4)\r<BR>\n" );
document.write( "\n" );
document.write( "x^2+x+4x+4=0 ..........Replace 5x with x+4x\r<BR>\n" );
document.write( "\n" );
document.write( "Group the first two terms together and the last two terms together like this:\r<BR>\n" );
document.write( "\n" );
document.write( "(x^2+x)+(4x+4)=0\r<BR>\n" );
document.write( "\n" );
document.write( "Factor a 1x out of the first group and factor a 4 out of the second group.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x(x+1)+4(x+1)=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now since we have a common term x+1 we can combine the two terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(x+4)(x+1)=0 ..........Combine like terms.<BR>\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the quadratic x^2+5x+4=0 factors to (x+4)(x+1)=0<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );