document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=146173>Question 146173</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0131492624/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Factor and check:  x^2 - 3xy - 4y^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #106722 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=106722\">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=x%5E2-3xy-4y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-3xy-4y%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=x%5E2 ALIGN=MIDDLE  ALT=\"x%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=-4y%5E2 ALIGN=MIDDLE  ALT=\"-4y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 1 and -4 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 1 and the last coefficient -4 to get -4. Now what two numbers multiply to -4 and add to the  middle coefficient -3? Let's list all of the factors of -4:\r<BR>\n" );
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document.write( "Factors of -4:\r<BR>\n" );
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document.write( "1,2\r<BR>\n" );
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document.write( "-1,-2 ...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 -4\r<BR>\n" );
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document.write( "(1)*(-4)\r<BR>\n" );
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document.write( "(-1)*(4)\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\">-4</TD><TD >1+(-4)=-3</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">4</TD><TD >-1+4=3</TD></tr>\n" );
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document.write( "From this list we can see that 1 and -4 add up to -3 and multiply to -4\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=x%5E2-3xy-4y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-3xy-4y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3xy ALIGN=MIDDLE  ALT=\"-3xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%2B-4xy ALIGN=MIDDLE  ALT=\"xy%2B-4xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%2B-4xy ALIGN=MIDDLE  ALT=\"xy%2B-4xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3xy ALIGN=MIDDLE  ALT=\"-3xy\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3xy ALIGN=MIDDLE  ALT=\"-3xy\"   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=x%5E2%2Bhighlight%28xy%2B-4xy%29%2B-4y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2Bhighlight%28xy%2B-4xy%29%2B-4y%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=x%5E2%2Bxy-4xy-4y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2Bxy-4xy-4y%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=%28x%5E2%2Bxy%29%2B%28-4xy-4y%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2Bxy%29%2B%28-4xy-4y%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%28x%2By%29-4y%28x%2By%29 ALIGN=MIDDLE  ALT=\"x%28x%2By%29-4y%28x%2By%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=-4y ALIGN=MIDDLE  ALT=\"-4y\"   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-4y%29%28x%2By%29 ALIGN=MIDDLE  ALT=\"%28x-4y%29%28x%2By%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=x%2By ALIGN=MIDDLE  ALT=\"x%2By\"   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=x%5E2%2Bxy-4xy-4y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2Bxy-4xy-4y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4y%29%28x%2By%29 ALIGN=MIDDLE  ALT=\"%28x-4y%29%28x%2By%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=x%5E2-3xy-4y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-3xy-4y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4y%29%28x%2By%29 ALIGN=MIDDLE  ALT=\"%28x-4y%29%28x%2By%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3xy-4y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-3xy-4y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bxy-4xy-4y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2Bxy-4xy-4y%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=x%5E2-3xy-4y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-3xy-4y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4y%29%28x%2By%29 ALIGN=MIDDLE  ALT=\"%28x-4y%29%28x%2By%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "In order to check our answer, we need to FOIL  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4y%29%28x%2By%29 ALIGN=MIDDLE  ALT=\"%28x-4y%29%28x%2By%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Remember, when you FOIL an expression, you follow this procedure:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28highlight%28x%29-4y%29%28highlight%28x%29%2By%29 ALIGN=MIDDLE  ALT=\"%28highlight%28x%29-4y%29%28highlight%28x%29%2By%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the <font color=\"red\">F</font>irst terms:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29%2A%28x%29=x%5E2 ALIGN=MIDDLE  ALT=\"%28x%29%2A%28x%29=x%5E2\"   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=%28highlight%28x%29-4y%29%28x%2Bhighlight%28y%29%29 ALIGN=MIDDLE  ALT=\"%28highlight%28x%29-4y%29%28x%2Bhighlight%28y%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the <font color=\"red\">O</font>uter terms:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29%2A%28y%29=x%2Ay ALIGN=MIDDLE  ALT=\"%28x%29%2A%28y%29=x%2Ay\"   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=%28x%2Bhighlight%28-4y%29%29%28highlight%28x%29%2By%29 ALIGN=MIDDLE  ALT=\"%28x%2Bhighlight%28-4y%29%29%28highlight%28x%29%2By%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the <font color=\"red\">I</font>nner terms:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-4%2Ay%29%2A%28x%29=-4%2Ax%2Ay ALIGN=MIDDLE  ALT=\"%28-4%2Ay%29%2A%28x%29=-4%2Ax%2Ay\"   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=%28x%2Bhighlight%28-4y%29%29%28x%2Bhighlight%28y%29%29 ALIGN=MIDDLE  ALT=\"%28x%2Bhighlight%28-4y%29%29%28x%2Bhighlight%28y%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the <font color=\"red\">L</font>ast terms:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-4%2Ay%29%2A%28y%29=-4%2Ay%5E2 ALIGN=MIDDLE  ALT=\"%28-4%2Ay%29%2A%28y%29=-4%2Ay%5E2\"   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=x%5E2%2Bx%2Ay-4%2Ax%2Ay-4%2Ay%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx%2Ay-4%2Ax%2Ay-4%2Ay%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now collect every term to make a single expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3%2Ax%2Ay-4%2Ay%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-3%2Ax%2Ay-4%2Ay%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now 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=%28x-4y%29%28x%2By%29 ALIGN=MIDDLE  ALT=\"%28x-4y%29%28x%2By%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOILS to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3%2Ax%2Ay-4%2Ay%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-3%2Ax%2Ay-4%2Ay%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4y%29%28x%2By%29=x%5E2-3%2Ax%2Ay-4%2Ay%5E2 ALIGN=MIDDLE  ALT=\"%28x-4y%29%28x%2By%29=x%5E2-3%2Ax%2Ay-4%2Ay%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So this verifies our answer.<BR>\n" );
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