document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=273700>Question 273700</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help with the following problem, including the steps needed to get to the answer. Thanks so much!!\r<BR>\n" );
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document.write( "Find the value of c to make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%5E2-18g%2Bc ALIGN=MIDDLE  ALT=\"g%5E2-18g%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > a perfect square. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #199907 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=199907\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's the quick way to do it:\r<BR>\n" );
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document.write( "Step 1) Take half of the 'g' coefficient -18 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18%2F2=-9 ALIGN=MIDDLE  ALT=\"-18%2F2=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 2) Square -9 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-9%29%5E2=81 ALIGN=MIDDLE  ALT=\"%28-9%29%5E2=81\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the value of 'c' is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=81 ALIGN=MIDDLE  ALT=\"c=81\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%5E2-18g%2B81 ALIGN=MIDDLE  ALT=\"g%5E2-18g%2B81\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here's why it works:\r<BR>\n" );
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document.write( "Since you want <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%5E2-18g%2Bc ALIGN=MIDDLE  ALT=\"g%5E2-18g%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be a perfect square, this means you want <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%5E2-18g%2Bc ALIGN=MIDDLE  ALT=\"g%5E2-18g%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28g%2Ba%29%5E2 ALIGN=MIDDLE  ALT=\"%28g%2Ba%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where 'a' is some fixed number. In other words, you want <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%5E2-18g%2Bc=%28g%2Ba%29%5E2 ALIGN=MIDDLE  ALT=\"g%5E2-18g%2Bc=%28g%2Ba%29%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=g%5E2-18g%2Bc=%28g%2Ba%29%5E2 ALIGN=MIDDLE  ALT=\"g%5E2-18g%2Bc=%28g%2Ba%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%5E2-18g%2Bc=g%5E2%2B2ag%2Ba%5E2 ALIGN=MIDDLE  ALT=\"g%5E2-18g%2Bc=g%5E2%2B2ag%2Ba%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "Take note that the 'g' term on the right side is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2ag ALIGN=MIDDLE  ALT=\"2ag\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18g=2ag ALIGN=MIDDLE  ALT=\"-18g=2ag\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Solve for 'a' to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-9 ALIGN=MIDDLE  ALT=\"a=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: in the step above, notice how we divided by 2. This is why we divided by 2 in the first step in the \"quick method\" shown above.\r<BR>\n" );
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document.write( "Also, notice that the constant term (the term without a 'g' in it) on the right is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2 ALIGN=MIDDLE  ALT=\"a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=a%5E2 ALIGN=MIDDLE  ALT=\"c=a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. But since we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-9 ALIGN=MIDDLE  ALT=\"a=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=%28-9%29%5E2=81 ALIGN=MIDDLE  ALT=\"c=%28-9%29%5E2=81\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Note: this explains why we squared the halved value of the middle coefficient.\r<BR>\n" );
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document.write( "So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=81 ALIGN=MIDDLE  ALT=\"c=81\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving us the polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%5E2-18g%2B81 ALIGN=MIDDLE  ALT=\"g%5E2-18g%2B81\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is a perfect square since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%5E2-18g%2B81=%28g-9%29%5E2 ALIGN=MIDDLE  ALT=\"g%5E2-18g%2B81=%28g-9%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  </SPAN>\n" );
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