document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=407874>Question 407874</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello, thank you much for taking your time to help me!  It's much appreciated. \r<BR>\n" );
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document.write( "I need help factoring an equation.<BR>\n" );
document.write( "I know the original equation and I know it's factored answer.<BR>\n" );
document.write( "The part that I can't understand is how to get there.\r<BR>\n" );
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document.write( "The equation is:   6a^2 - 9ab - 15b^2\r<BR>\n" );
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document.write( "The answer is:     3(2a - 5b)(a + b)\r<BR>\n" );
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document.write( "When I factor it, I reach:  3(2a^2 - 3ab - 5b^2)\r<BR>\n" );
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document.write( "Where do I go from here?  How do I do it?  Why do I do it that way?\r<BR>\n" );
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document.write( "Any help is beyond loved!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #287398 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=287398\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6a%5E2-9ab-15b%5E2 ALIGN=MIDDLE  ALT=\"6a%5E2-9ab-15b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%282a%5E2-3ab-5b%5E2%29 ALIGN=MIDDLE  ALT=\"3%282a%5E2-3ab-5b%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's focus on the inner expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2-3ab-5b%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2-3ab-5b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2-3ab-5b%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2-3ab-5b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5b%5E2 ALIGN=MIDDLE  ALT=\"-5b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 2 and -5 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 2 and the last coefficient -5 to get -10. Now what two numbers multiply to -10 and add to the  middle coefficient -3? Let's list all of the factors of -10:\r<BR>\n" );
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document.write( "Factors of -10:\r<BR>\n" );
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document.write( "1,2,5,10\r<BR>\n" );
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document.write( "-1,-2,-5,-10 ...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 and multiply to -10\r<BR>\n" );
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document.write( "(1)*(-10)\r<BR>\n" );
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document.write( "(2)*(-5)\r<BR>\n" );
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document.write( "(-1)*(10)\r<BR>\n" );
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document.write( "(-2)*(5)\r<BR>\n" );
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document.write( "note: remember, the product of a negative and a positive number is a negative number\r<BR>\n" );
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document.write( "Now which of these pairs add to -3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -3\r<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD  align=\"center\">1</TD><TD  align=\"center\">-10</TD><TD >1+(-10)=-9</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">-5</TD><TD >2+(-5)=-3</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">10</TD><TD >-1+10=9</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">5</TD><TD >-2+5=3</TD></tr>\n" );
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document.write( "From this list we can see that 2 and -5 add up to -3 and multiply to -10\r<BR>\n" );
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document.write( "Now looking at the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2-3ab-5b%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2-3ab-5b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3ab ALIGN=MIDDLE  ALT=\"-3ab\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2ab%2B-5ab ALIGN=MIDDLE  ALT=\"2ab%2B-5ab\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2ab%2B-5ab ALIGN=MIDDLE  ALT=\"2ab%2B-5ab\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3ab ALIGN=MIDDLE  ALT=\"-3ab\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3ab ALIGN=MIDDLE  ALT=\"-3ab\"   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=2a%5E2%2Bhighlight%282ab%2B-5ab%29%2B-5b%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2%2Bhighlight%282ab%2B-5ab%29%2B-5b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2%2B2ab-5ab-5b%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2%2B2ab-5ab-5b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > by grouping:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a%5E2%2B2ab%29%2B%28-5ab-5b%5E2%29 ALIGN=MIDDLE  ALT=\"%282a%5E2%2B2ab%29%2B%28-5ab-5b%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%28a%2Bb%29-5b%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"2a%28a%2Bb%29-5b%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a ALIGN=MIDDLE  ALT=\"2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the first group. Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5b ALIGN=MIDDLE  ALT=\"-5b\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the second group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a-5b%29%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"%282a-5b%29%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since we have a common term of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb ALIGN=MIDDLE  ALT=\"a%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can combine like terms\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2%2B2ab-5ab-5b%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2%2B2ab-5ab-5b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a-5b%29%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"%282a-5b%29%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this also means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2-3ab-5b%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2-3ab-5b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a-5b%29%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"%282a-5b%29%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2-3ab-5b%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2-3ab-5b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2%2B2ab-5ab-5b%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2%2B2ab-5ab-5b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "So our expression goes from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%282a%5E2-3ab-5b%5E2%29 ALIGN=MIDDLE  ALT=\"3%282a%5E2-3ab-5b%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and factors further to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%282a-5b%29%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"3%282a-5b%29%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6a%5E2-9ab-15b%5E2 ALIGN=MIDDLE  ALT=\"6a%5E2-9ab-15b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%282a-5b%29%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"3%282a-5b%29%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "    \r<BR>\n" );
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document.write( "If you need more help, email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com\">jim_thompson5910@hotmail.com</a>\r<BR>\n" );
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document.write( "Jim </SPAN>\n" );
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