document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=77522>Question 77522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please show me how to SOLVE my homework question. thanks. \r<BR>\n" );
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document.write( "Y^2-36/y^2-4 multiplied by y^2+6y+8/y^2-2y-24 \r<BR>\n" );
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document.write( "the choices are: \r<BR>\n" );
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document.write( "A.) y+6/y-2 B.) y-6/y+2 c.) y+2/y+6 D.) y-2/y-6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #55611 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=55611\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28y%5E2-36%29%2F%28y%5E2-4%29%29%2A%28%28y%5E2%2B6y%2B8%29%2F%28y%5E2-2y-24%29%29 ALIGN=MIDDLE  ALT=\"%28%28y%5E2-36%29%2F%28y%5E2-4%29%29%2A%28%28y%5E2%2B6y%2B8%29%2F%28y%5E2-2y-24%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Factor the first numerator\r<BR>\n" );
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document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=factoring_quadratics.solver>Factoring Quadratics with a leading coefficient of 1 (a=1)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>In order to factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%5E2%2B0%2Ax%2B-36 ALIGN=MIDDLE  ALT=\"1%2Ax%5E2%2B0%2Ax%2B-36\"   DISABLED_style= \"max-width:90%; height:auto;\" >, first 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<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Factors:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  1,2,3,4,6,9,12,18,36,<BR>\n" );
document.write( "  <BR>\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<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  These factors pair up to multiply to -36.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-1)*(36)=-36<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-2)*(18)=-36<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-3)*(12)=-36<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-4)*(9)=-36<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-6)*(6)=-36<BR>\n" );
document.write( "  <BR>\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<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <TABLE><TR><TD>First Number</TD><TD >|</TD><TD >Second Number</TD><TD >|</TD><TD >Sum</TD></TR><TR><TD >1</TD><TD >|</TD><TD >-36</TD>|</TD><TD >|</TD><TD >1+(-36)=-35</TD></TR><TR><TD >2</TD><TD >|</TD><TD >-18</TD>|</TD><TD >|</TD><TD >2+(-18)=-16</TD></TR><TR><TD >3</TD><TD >|</TD><TD >-12</TD>|</TD><TD >|</TD><TD >3+(-12)=-9</TD></TR><TR><TD >4</TD><TD >|</TD><TD >-9</TD>|</TD><TD >|</TD><TD >4+(-9)=-5</TD></TR><TR><TD >6</TD><TD >|</TD><TD >-6</TD>|</TD><TD >|</TD><TD >6+(-6)=0</TD></TR><TR><TD >-1</TD><TD >|</TD><TD >36</TD>|</TD><TD >|</TD><TD >(-1)+36=35</TD></TR><TR><TD >-2</TD><TD >|</TD><TD >18</TD>|</TD><TD >|</TD><TD >(-2)+18=16</TD></TR><TR><TD >-3</TD><TD >|</TD><TD >12</TD>|</TD><TD >|</TD><TD >(-3)+12=9</TD></TR><TR><TD >-4</TD><TD >|</TD><TD >9</TD>|</TD><TD >|</TD><TD >(-4)+9=5</TD></TR><TR><TD >-6</TD><TD >|</TD><TD >6</TD>|</TD><TD >|</TD><TD >(-6)+6=0</TD></TR><TABLE>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\n" );
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document.write( "  Now we substitute these numbers into a and b of the general equation of a product of linear factors which is:\r\n" );
document.write( "  \r\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Ba%29%28x%2Bb%29 ALIGN=MIDDLE  ALT=\"%28x%2Ba%29%28x%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >substitute a=-6 and b=6\r\n" );
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document.write( "  So the equation becomes:\r\n" );
document.write( "  \r\n" );
document.write( "  (x-6)(x+6)\r\n" );
document.write( "  \r\n" );
document.write( "  Notice that if we foil (x-6)(x+6) we get the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%5E2%2B0%2Ax%2B-36 ALIGN=MIDDLE  ALT=\"1%2Ax%5E2%2B0%2Ax%2B-36\"   DISABLED_style= \"max-width:90%; height:auto;\" > again</TD></TR>\n" );
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document.write( "Factor the first denominator\r<BR>\n" );
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document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=factoring_quadratics.solver>Factoring Quadratics with a leading coefficient of 1 (a=1)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>In order to factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%5E2%2B0%2Ax%2B-4 ALIGN=MIDDLE  ALT=\"1%2Ax%5E2%2B0%2Ax%2B-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, first we need to ask ourselves: What two numbers multiply to -4 and add to 0? Lets find out by listing all of the possible factors of -4<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Factors:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  1,2,4,4,6,9,12,18,36,<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  -1,-2,-4,-4,-6,-9,-12,-18,-36,List the negative factors as well. This will allow us to find all possible combinations<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  These factors pair up to multiply to -4.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-1)*(36)=-4<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-2)*(18)=-4<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-4)*(12)=-4<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-4)*(9)=-4<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-6)*(6)=-4<BR>\n" );
document.write( "  <BR>\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<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <TABLE><TR><TD>First Number</TD><TD >|</TD><TD >Second Number</TD><TD >|</TD><TD >Sum</TD></TR><TR><TD >1</TD><TD >|</TD><TD >-36</TD>|</TD><TD >|</TD><TD >1+(-36)=-35</TD></TR><TR><TD >2</TD><TD >|</TD><TD >-18</TD>|</TD><TD >|</TD><TD >2+(-18)=-16</TD></TR><TR><TD >4</TD><TD >|</TD><TD >-12</TD>|</TD><TD >|</TD><TD >4+(-12)=-8</TD></TR><TR><TD >4</TD><TD >|</TD><TD >-9</TD>|</TD><TD >|</TD><TD >4+(-9)=-5</TD></TR><TR><TD >6</TD><TD >|</TD><TD >-6</TD>|</TD><TD >|</TD><TD >6+(-6)=0</TD></TR><TR><TD >-1</TD><TD >|</TD><TD >36</TD>|</TD><TD >|</TD><TD >(-1)+36=35</TD></TR><TR><TD >-2</TD><TD >|</TD><TD >18</TD>|</TD><TD >|</TD><TD >(-2)+18=16</TD></TR><TR><TD >-4</TD><TD >|</TD><TD >12</TD>|</TD><TD >|</TD><TD >(-4)+12=8</TD></TR><TR><TD >-4</TD><TD >|</TD><TD >9</TD>|</TD><TD >|</TD><TD >(-4)+9=5</TD></TR><TR><TD >-6</TD><TD >|</TD><TD >6</TD>|</TD><TD >|</TD><TD >(-6)+6=0</TD></TR><TABLE>We can see from the table that -6 and 6 add to 0.So the two numbers that multiply to -4 and add to 0 are: -6 and 6\r\n" );
document.write( "  \r\n" );
document.write( "  Now we substitute these numbers into a and b of the general equation of a product of linear factors which is:\r\n" );
document.write( "  \r\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Ba%29%28x%2Bb%29 ALIGN=MIDDLE  ALT=\"%28x%2Ba%29%28x%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >substitute a=-6 and b=6\r\n" );
document.write( "  \r\n" );
document.write( "  So the equation becomes:\r\n" );
document.write( "  \r\n" );
document.write( "  (x-6)(x+6)\r\n" );
document.write( "  \r\n" );
document.write( "  Notice that if we foil (x-6)(x+6) we get the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%5E2%2B0%2Ax%2B-4 ALIGN=MIDDLE  ALT=\"1%2Ax%5E2%2B0%2Ax%2B-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > again</TD></TR>\n" );
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document.write( "Factor the second numerator\r<BR>\n" );
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document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=factoring_quadratics.solver>Factoring Quadratics with a leading coefficient of 1 (a=1)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>In order to factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%5E2%2B6%2Ax%2B8 ALIGN=MIDDLE  ALT=\"1%2Ax%5E2%2B6%2Ax%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, first we need to ask ourselves: What two numbers multiply to 8 and add to 6? Lets find out by listing all of the possible factors of 8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Factors:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  1,2,4,8,6,9,12,18,36,<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  -1,-2,-4,-8,-6,-9,-12,-18,-36,List the negative factors as well. This will allow us to find all possible combinations<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  These factors pair up to multiply to 8.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  1*36=8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  2*18=8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  4*12=8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  8*9=8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  6*6=8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-1)*(-36)=8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-2)*(-18)=8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-4)*(-12)=8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-8)*(-9)=8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-6)*(-6)=8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  note: remember two negative numbers multiplied together make a positive number<BR>\n" );
document.write( "  <BR>\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<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <TABLE><TR><TD>First Number</TD><TD >|</TD><TD >Second Number</TD><TD >|</TD><TD >Sum</TD></TR><TR><TD >1</TD><TD >|</TD><TD >36</TD>|</TD><TD >|</TD><TD >1+36=37</TD></TR><TR><TD >2</TD><TD >|</TD><TD >18</TD>|</TD><TD >|</TD><TD >2+18=20</TD></TR><TR><TD >4</TD><TD >|</TD><TD >12</TD>|</TD><TD >|</TD><TD >4+12=16</TD></TR><TR><TD >8</TD><TD >|</TD><TD >9</TD>|</TD><TD >|</TD><TD >8+9=17</TD></TR><TR><TD >6</TD><TD >|</TD><TD >6</TD>|</TD><TD >|</TD><TD >6+6=12</TD></TR><TR><TD >-1</TD><TD >|</TD><TD >-36</TD>|</TD><TD >|</TD><TD >-1+(-36)=-37</TD></TR><TR><TD >-2</TD><TD >|</TD><TD >-18</TD>|</TD><TD >|</TD><TD >-2+(-18)=-20</TD></TR><TR><TD >-4</TD><TD >|</TD><TD >-12</TD>|</TD><TD >|</TD><TD >-4+(-12)=-16</TD></TR><TR><TD >-8</TD><TD >|</TD><TD >-9</TD>|</TD><TD >|</TD><TD >-8+(-9)=-17</TD></TR><TR><TD >-6</TD><TD >|</TD><TD >-6</TD>|</TD><TD >|</TD><TD >-6+(-6)=-12</TD></TR><TABLE>substitute a=-6 and b=6\r\n" );
document.write( "  \r\n" );
document.write( "  So the equation becomes:\r\n" );
document.write( "  \r\n" );
document.write( "  (x-6)(x+6)\r\n" );
document.write( "  \r\n" );
document.write( "  Notice that if we foil (x-6)(x+6) we get the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%5E2%2B6%2Ax%2B8 ALIGN=MIDDLE  ALT=\"1%2Ax%5E2%2B6%2Ax%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > again\r\n" );
document.write( "  \r\n" );
document.write( "  None of these factors add to 6. So this quadratic cannot be factored. In order to solve for x, we need to use the quadratic formula.</TD></TR>\n" );
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document.write( "Factor the second denominator\r<BR>\n" );
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document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=factoring_quadratics.solver>Factoring Quadratics with a leading coefficient of 1 (a=1)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>In order to factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%5E2%2B-2%2Ax%2B-24 ALIGN=MIDDLE  ALT=\"1%2Ax%5E2%2B-2%2Ax%2B-24\"   DISABLED_style= \"max-width:90%; height:auto;\" >, first we need to ask ourselves: What two numbers multiply to -24 and add to -2? Lets find out by listing all of the possible factors of -24<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Factors:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  1,2,3,4,6,8,12,24,36,<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  -1,-2,-3,-4,-6,-8,-12,-24,-36,List the negative factors as well. This will allow us to find all possible combinations<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  These factors pair up to multiply to -24.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-1)*(36)=-24<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-2)*(24)=-24<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-3)*(12)=-24<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-4)*(8)=-24<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (-6)*(6)=-24<BR>\n" );
document.write( "  <BR>\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<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <TABLE><TR><TD>First Number</TD><TD >|</TD><TD >Second Number</TD><TD >|</TD><TD >Sum</TD></TR><TR><TD >1</TD><TD >|</TD><TD >-36</TD>|</TD><TD >|</TD><TD >1+(-36)=-35</TD></TR><TR><TD >2</TD><TD >|</TD><TD >-24</TD>|</TD><TD >|</TD><TD >2+(-24)=-22</TD></TR><TR><TD >3</TD><TD >|</TD><TD >-12</TD>|</TD><TD >|</TD><TD >3+(-12)=-9</TD></TR><TR><TD >4</TD><TD >|</TD><TD >-8</TD>|</TD><TD >|</TD><TD >4+(-8)=-4</TD></TR><TR><TD >6</TD><TD >|</TD><TD >-6</TD>|</TD><TD >|</TD><TD >6+(-6)=0</TD></TR><TR><TD >-1</TD><TD >|</TD><TD >36</TD>|</TD><TD >|</TD><TD >(-1)+36=35</TD></TR><TR><TD >-2</TD><TD >|</TD><TD >24</TD>|</TD><TD >|</TD><TD >(-2)+24=22</TD></TR><TR><TD >-3</TD><TD >|</TD><TD >12</TD>|</TD><TD >|</TD><TD >(-3)+12=9</TD></TR><TR><TD >-4</TD><TD >|</TD><TD >8</TD>|</TD><TD >|</TD><TD >(-4)+8=4</TD></TR><TR><TD >-6</TD><TD >|</TD><TD >6</TD>|</TD><TD >|</TD><TD >(-6)+6=0</TD></TR><TABLE>substitute a=-6 and b=6\r\n" );
document.write( "  \r\n" );
document.write( "  So the equation becomes:\r\n" );
document.write( "  \r\n" );
document.write( "  (x-6)(x+6)\r\n" );
document.write( "  \r\n" );
document.write( "  Notice that if we foil (x-6)(x+6) we get the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%5E2%2B-2%2Ax%2B-24 ALIGN=MIDDLE  ALT=\"1%2Ax%5E2%2B-2%2Ax%2B-24\"   DISABLED_style= \"max-width:90%; height:auto;\" > again\r\n" );
document.write( "  \r\n" );
document.write( "  None of these factors add to -2. So this quadratic cannot be factored. In order to solve for x, we need to use the quadratic formula.</TD></TR>\n" );
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document.write( "So the whole expression becomes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28%28y-6%29%28y%2B6%29%29%2F%28%28y-2%29%28y%2B2%29%29%29%2A%28%28%28y%2B2%29%28y%2B4%29%29%2F%28%28y%2B4%29%28y-6%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28cross%28%28y-6%29%29%28y%2B6%29%29%2F%28%28y-2%29cross%28%28y%2B2%29%29%29%29%2A%28%28cross%28%28y%2B2%29%29cross%28%28y%2B4%29%29%29%2F%28cross%28%28y%2B4%29%29cross%28%28y-6%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  Cancel like terms\r<BR>\n" );
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document.write( "So the whole expression reduces to \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y%2B6%29%2F%28y-2%29 ALIGN=MIDDLE  ALT=\"%28y%2B6%29%2F%28y-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "So the answer is A) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );