document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=147362>Question 147362</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor; 4a^2-4a+1 <BR>\n" );
document.write( "I did it just like by this problem; a^2-9= a^2-3^2= (a+3)(a-3, but it was wrong.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #107734 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=107734\">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=4a%5E2-4a%2B1 ALIGN=MIDDLE  ALT=\"4a%5E2-4a%2B1\"   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=4a%5E2 ALIGN=MIDDLE  ALT=\"4a%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=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 4 and 1 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 4 and the last coefficient 1 to get 4. Now what two numbers multiply to 4 and add to the  middle coefficient -4? 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( "2*2\r<BR>\n" );
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document.write( "(-1)*(-4)\r<BR>\n" );
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document.write( "(-2)*(-2)\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 -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( "<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=5</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">2</TD><TD >2+2=4</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-4</TD><TD >-1+(-4)=-5</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">-2</TD><TD >-2+(-2)=-4</TD></tr>\n" );
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document.write( "From this list we can see that -2 and -2 add up to -4 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=4a%5E2-4a%2B1 ALIGN=MIDDLE  ALT=\"4a%5E2-4a%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4a ALIGN=MIDDLE  ALT=\"-4a\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2a%2B-2a ALIGN=MIDDLE  ALT=\"-2a%2B-2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2a%2B-2a ALIGN=MIDDLE  ALT=\"-2a%2B-2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4a ALIGN=MIDDLE  ALT=\"-4a\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4a ALIGN=MIDDLE  ALT=\"-4a\"   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=4a%5E2%2Bhighlight%28-2a%2B-2a%29%2B1 ALIGN=MIDDLE  ALT=\"4a%5E2%2Bhighlight%28-2a%2B-2a%29%2B1\"   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=4a%5E2-2a-2a%2B1 ALIGN=MIDDLE  ALT=\"4a%5E2-2a-2a%2B1\"   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=%284a%5E2-2a%29%2B%28-2a%2B1%29 ALIGN=MIDDLE  ALT=\"%284a%5E2-2a%29%2B%28-2a%2B1%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%282a-1%29-1%282a-1%29 ALIGN=MIDDLE  ALT=\"2a%282a-1%29-1%282a-1%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=-1 ALIGN=MIDDLE  ALT=\"-1\"   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-1%29%282a-1%29 ALIGN=MIDDLE  ALT=\"%282a-1%29%282a-1%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=2a-1 ALIGN=MIDDLE  ALT=\"2a-1\"   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=4a%5E2-2a-2a%2B1 ALIGN=MIDDLE  ALT=\"4a%5E2-2a-2a%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a-1%29%282a-1%29 ALIGN=MIDDLE  ALT=\"%282a-1%29%282a-1%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=4a%5E2-4a%2B1 ALIGN=MIDDLE  ALT=\"4a%5E2-4a%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a-1%29%282a-1%29 ALIGN=MIDDLE  ALT=\"%282a-1%29%282a-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%5E2-4a%2B1 ALIGN=MIDDLE  ALT=\"4a%5E2-4a%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%5E2-2a-2a%2B1 ALIGN=MIDDLE  ALT=\"4a%5E2-2a-2a%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "note:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a-1%29%282a-1%29 ALIGN=MIDDLE  ALT=\"%282a-1%29%282a-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a-1%29%5E2 ALIGN=MIDDLE  ALT=\"%282a-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > since the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a-1 ALIGN=MIDDLE  ALT=\"2a-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > occurs twice. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%5E2-4a%2B1 ALIGN=MIDDLE  ALT=\"4a%5E2-4a%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > also factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a-1%29%5E2 ALIGN=MIDDLE  ALT=\"%282a-1%29%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=4a%5E2-4a%2B1 ALIGN=MIDDLE  ALT=\"4a%5E2-4a%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a-1%29%5E2 ALIGN=MIDDLE  ALT=\"%282a-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
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