document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=352211>Question 352211</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor completely.<BR>\n" );
document.write( "20w^2+100wg+125g^2\r<BR>\n" );
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document.write( "All of the examples I have offer numbers that have square roots so i'm not sure how to solve this one. Any help you can give is very much appreciated.\r<BR>\n" );
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document.write( "Thank you in advance! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #251687 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=251687\">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=20w%5E2%2B100wg%2B125g%5E2 ALIGN=MIDDLE  ALT=\"20w%5E2%2B100wg%2B125g%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=5%284w%5E2%2B20wg%2B25g%5E2%29 ALIGN=MIDDLE  ALT=\"5%284w%5E2%2B20wg%2B25g%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=5 ALIGN=MIDDLE  ALT=\"5\"   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=4w%5E2%2B20wg%2B25g%5E2 ALIGN=MIDDLE  ALT=\"4w%5E2%2B20wg%2B25g%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=4w%5E2%2B20wg%2B25g%5E2 ALIGN=MIDDLE  ALT=\"4w%5E2%2B20wg%2B25g%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=4w%5E2 ALIGN=MIDDLE  ALT=\"4w%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=25g%5E2 ALIGN=MIDDLE  ALT=\"25g%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 4 and 25 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 4 and the last coefficient 25 to get 100. Now what two numbers multiply to 100 and add to the  middle coefficient 20? Let's list all of the factors of 100:\r<BR>\n" );
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document.write( "Factors of 100:\r<BR>\n" );
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document.write( "1,2,4,5,10,20,25,50\r<BR>\n" );
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document.write( "-1,-2,-4,-5,-10,-20,-25,-50 ...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 100\r<BR>\n" );
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document.write( "1*100\r<BR>\n" );
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document.write( "2*50\r<BR>\n" );
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document.write( "4*25\r<BR>\n" );
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document.write( "5*20\r<BR>\n" );
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document.write( "10*10\r<BR>\n" );
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document.write( "(-1)*(-100)\r<BR>\n" );
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document.write( "(-2)*(-50)\r<BR>\n" );
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document.write( "(-4)*(-25)\r<BR>\n" );
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document.write( "(-5)*(-20)\r<BR>\n" );
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document.write( "(-10)*(-10)\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 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20\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\">100</TD><TD >1+100=101</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">50</TD><TD >2+50=52</TD></tr><tr><TD  align=\"center\">4</TD><TD  align=\"center\">25</TD><TD >4+25=29</TD></tr><tr><TD  align=\"center\">5</TD><TD  align=\"center\">20</TD><TD >5+20=25</TD></tr><tr><TD  align=\"center\">10</TD><TD  align=\"center\">10</TD><TD >10+10=20</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-100</TD><TD >-1+(-100)=-101</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">-50</TD><TD >-2+(-50)=-52</TD></tr><tr><TD  align=\"center\">-4</TD><TD  align=\"center\">-25</TD><TD >-4+(-25)=-29</TD></tr><tr><TD  align=\"center\">-5</TD><TD  align=\"center\">-20</TD><TD >-5+(-20)=-25</TD></tr><tr><TD  align=\"center\">-10</TD><TD  align=\"center\">-10</TD><TD >-10+(-10)=-20</TD></tr>\n" );
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document.write( "From this list we can see that 10 and 10 add up to 20 and multiply to 100\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=4w%5E2%2B20wg%2B25g%5E2 ALIGN=MIDDLE  ALT=\"4w%5E2%2B20wg%2B25g%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20wg ALIGN=MIDDLE  ALT=\"20wg\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10wg%2B10wg ALIGN=MIDDLE  ALT=\"10wg%2B10wg\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10wg%2B10wg ALIGN=MIDDLE  ALT=\"10wg%2B10wg\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20wg ALIGN=MIDDLE  ALT=\"20wg\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20wg ALIGN=MIDDLE  ALT=\"20wg\"   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=4w%5E2%2Bhighlight%2810wg%2B10wg%29%2B25g%5E2 ALIGN=MIDDLE  ALT=\"4w%5E2%2Bhighlight%2810wg%2B10wg%29%2B25g%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=4w%5E2%2B10wg%2B10wg%2B25g%5E2 ALIGN=MIDDLE  ALT=\"4w%5E2%2B10wg%2B10wg%2B25g%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=%284w%5E2%2B10wg%29%2B%2810wg%2B25g%5E2%29 ALIGN=MIDDLE  ALT=\"%284w%5E2%2B10wg%29%2B%2810wg%2B25g%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=2w%282w%2B5g%29%2B5g%282w%2B5g%29 ALIGN=MIDDLE  ALT=\"2w%282w%2B5g%29%2B5g%282w%2B5g%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=2w ALIGN=MIDDLE  ALT=\"2w\"   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=5g ALIGN=MIDDLE  ALT=\"5g\"   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=%282w%2B5g%29%282w%2B5g%29 ALIGN=MIDDLE  ALT=\"%282w%2B5g%29%282w%2B5g%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=2w%2B5g ALIGN=MIDDLE  ALT=\"2w%2B5g\"   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=4w%5E2%2B10wg%2B10wg%2B25g%5E2 ALIGN=MIDDLE  ALT=\"4w%5E2%2B10wg%2B10wg%2B25g%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282w%2B5g%29%282w%2B5g%29 ALIGN=MIDDLE  ALT=\"%282w%2B5g%29%282w%2B5g%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=4w%5E2%2B20wg%2B25g%5E2 ALIGN=MIDDLE  ALT=\"4w%5E2%2B20wg%2B25g%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282w%2B5g%29%282w%2B5g%29 ALIGN=MIDDLE  ALT=\"%282w%2B5g%29%282w%2B5g%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4w%5E2%2B20wg%2B25g%5E2 ALIGN=MIDDLE  ALT=\"4w%5E2%2B20wg%2B25g%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4w%5E2%2B10wg%2B10wg%2B25g%5E2 ALIGN=MIDDLE  ALT=\"4w%5E2%2B10wg%2B10wg%2B25g%5E2\"   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=%282w%2B5g%29%282w%2B5g%29 ALIGN=MIDDLE  ALT=\"%282w%2B5g%29%282w%2B5g%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282w%2B5g%29%5E2 ALIGN=MIDDLE  ALT=\"%282w%2B5g%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > since the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2w%2B5g ALIGN=MIDDLE  ALT=\"2w%2B5g\"   DISABLED_style= \"max-width:90%; height:auto;\" > occurs twice. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4w%5E2%2B20wg%2B25g%5E2 ALIGN=MIDDLE  ALT=\"4w%5E2%2B20wg%2B25g%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > also factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282w%2B5g%29%5E2 ALIGN=MIDDLE  ALT=\"%282w%2B5g%29%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=5%284w%5E2%2B20wg%2B25g%5E2%29 ALIGN=MIDDLE  ALT=\"5%284w%5E2%2B20wg%2B25g%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=5%282w%2B5g%29%5E2 ALIGN=MIDDLE  ALT=\"5%282w%2B5g%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=20w%5E2%2B100wg%2B125g%5E2 ALIGN=MIDDLE  ALT=\"20w%5E2%2B100wg%2B125g%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%282w%2B5g%29%5E2 ALIGN=MIDDLE  ALT=\"5%282w%2B5g%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "     </SPAN>\n" );
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