document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=117828>Question 117828</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor completely.     6x^2–xy–5y^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #85823 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=85823\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2-1xy-5y%5E2 ALIGN=MIDDLE  ALT=\"6x%5E2-1xy-5y%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=6x%5E2 ALIGN=MIDDLE  ALT=\"6x%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=-5y%5E2 ALIGN=MIDDLE  ALT=\"-5y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 6 and -5 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 6 and the last coefficient -5 to get -30. Now what two numbers multiply to -30 and add to the  middle coefficient -1? Let's list all of the factors of -30:\r<BR>\n" );
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document.write( "Factors of -30:\r<BR>\n" );
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document.write( "1,2,3,5,6,10,15,30\r<BR>\n" );
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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" );
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document.write( "These factors pair up and multiply to -30\r<BR>\n" );
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document.write( "(1)*(-30)\r<BR>\n" );
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document.write( "(2)*(-15)\r<BR>\n" );
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document.write( "(3)*(-10)\r<BR>\n" );
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document.write( "(5)*(-6)\r<BR>\n" );
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document.write( "(-1)*(30)\r<BR>\n" );
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document.write( "(-2)*(15)\r<BR>\n" );
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document.write( "(-3)*(10)\r<BR>\n" );
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document.write( "(-5)*(6)\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 -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" );
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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\">-30</TD><TD >1+(-30)=-29</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">-15</TD><TD >2+(-15)=-13</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">-10</TD><TD >3+(-10)=-7</TD></tr><tr><TD  align=\"center\">5</TD><TD  align=\"center\">-6</TD><TD >5+(-6)=-1</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">30</TD><TD >-1+30=29</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">15</TD><TD >-2+15=13</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">10</TD><TD >-3+10=7</TD></tr><tr><TD  align=\"center\">-5</TD><TD  align=\"center\">6</TD><TD >-5+6=1</TD></tr>\n" );
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document.write( "From this list we can see that 5 and -6 add up to -1 and multiply to -30\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=6x%5E2-1xy-5y%5E2 ALIGN=MIDDLE  ALT=\"6x%5E2-1xy-5y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1xy ALIGN=MIDDLE  ALT=\"-1xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5xy%2B-6xy ALIGN=MIDDLE  ALT=\"5xy%2B-6xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5xy%2B-6xy ALIGN=MIDDLE  ALT=\"5xy%2B-6xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1xy ALIGN=MIDDLE  ALT=\"-1xy\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1xy ALIGN=MIDDLE  ALT=\"-1xy\"   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=6x%5E2%2Bhighlight%285xy%2B-6xy%29%2B-5y%5E2 ALIGN=MIDDLE  ALT=\"6x%5E2%2Bhighlight%285xy%2B-6xy%29%2B-5y%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=6x%5E2%2B5xy-6xy-5y%5E2 ALIGN=MIDDLE  ALT=\"6x%5E2%2B5xy-6xy-5y%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=%286x%5E2%2B5xy%29%2B%28-6xy-5y%5E2%29 ALIGN=MIDDLE  ALT=\"%286x%5E2%2B5xy%29%2B%28-6xy-5y%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=x%286x%2B5y%29-y%286x%2B5y%29 ALIGN=MIDDLE  ALT=\"x%286x%2B5y%29-y%286x%2B5y%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=x ALIGN=MIDDLE  ALT=\"x\"   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=-y ALIGN=MIDDLE  ALT=\"-y\"   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=%28x-y%29%286x%2B5y%29 ALIGN=MIDDLE  ALT=\"%28x-y%29%286x%2B5y%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=6x%2B5y ALIGN=MIDDLE  ALT=\"6x%2B5y\"   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=6x%5E2%2B5xy-6xy-5y%5E2 ALIGN=MIDDLE  ALT=\"6x%5E2%2B5xy-6xy-5y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-y%29%286x%2B5y%29 ALIGN=MIDDLE  ALT=\"%28x-y%29%286x%2B5y%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=6x%5E2-1xy-5y%5E2 ALIGN=MIDDLE  ALT=\"6x%5E2-1xy-5y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-y%29%286x%2B5y%29 ALIGN=MIDDLE  ALT=\"%28x-y%29%286x%2B5y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2-1xy-5y%5E2 ALIGN=MIDDLE  ALT=\"6x%5E2-1xy-5y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B5xy-6xy-5y%5E2 ALIGN=MIDDLE  ALT=\"6x%5E2%2B5xy-6xy-5y%5E2\"   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=6x%5E2-1xy-5y%5E2 ALIGN=MIDDLE  ALT=\"6x%5E2-1xy-5y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-y%29%286x%2B5y%29 ALIGN=MIDDLE  ALT=\"%28x-y%29%286x%2B5y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );