document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=656705>Question 656705</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 20a squared + 39a + 8 <BR>\n" );
document.write( "factor out completely </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #409580 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=409580\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=completing_the_square.solver>Completing the Square to Get a Quadratic into Vertex Form</A></TD></TR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=20+x%5E2%2B39+x%2B8 ALIGN=MIDDLE  ALT=\"y=20+x%5E2%2B39+x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8=20+x%5E2%2B39+x ALIGN=MIDDLE  ALT=\"y-8=20+x%5E2%2B39+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8=20%28x%5E2%2B%2839%2F20%29x%29 ALIGN=MIDDLE  ALT=\"y-8=20%28x%5E2%2B%2839%2F20%29x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the leading coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Take half of the x coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39%2F20 ALIGN=MIDDLE  ALT=\"39%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39%2F40 ALIGN=MIDDLE  ALT=\"39%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2839%2F20%29=39%2F40 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2839%2F20%29=39%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
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document.write( "  Now square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39%2F40 ALIGN=MIDDLE  ALT=\"39%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1521%2F1600 ALIGN=MIDDLE  ALT=\"1521%2F1600\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2839%2F40%29%5E2=%2839%2F40%29%2839%2F40%29=1521%2F1600 ALIGN=MIDDLE  ALT=\"%2839%2F40%29%5E2=%2839%2F40%29%2839%2F40%29=1521%2F1600\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8=20%28x%5E2%2B%2839%2F20%29x%2B1521%2F1600-1521%2F1600%29 ALIGN=MIDDLE  ALT=\"y-8=20%28x%5E2%2B%2839%2F20%29x%2B1521%2F1600-1521%2F1600%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1521%2F1600 ALIGN=MIDDLE  ALT=\"1521%2F1600\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not change the equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8=20%28%28x%2B39%2F40%29%5E2-1521%2F1600%29 ALIGN=MIDDLE  ALT=\"y-8=20%28%28x%2B39%2F40%29%5E2-1521%2F1600%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%2839%2F20%29x%2B1521%2F1600 ALIGN=MIDDLE  ALT=\"x%5E2%2B%2839%2F20%29x%2B1521%2F1600\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B39%2F40%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B39%2F40%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8=20%28x%2B39%2F40%29%5E2-20%281521%2F1600%29 ALIGN=MIDDLE  ALT=\"y-8=20%28x%2B39%2F40%29%5E2-20%281521%2F1600%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8=20%28x%2B39%2F40%29%5E2-1521%2F80 ALIGN=MIDDLE  ALT=\"y-8=20%28x%2B39%2F40%29%5E2-1521%2F80\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=20%28x%2B39%2F40%29%5E2-1521%2F80%2B8 ALIGN=MIDDLE  ALT=\"y=20%28x%2B39%2F40%29%5E2-1521%2F80%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides to isolate y<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=20%28x%2B39%2F40%29%5E2-881%2F80 ALIGN=MIDDLE  ALT=\"y=20%28x%2B39%2F40%29%5E2-881%2F80\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms<BR>\n" );
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document.write( "  Now the quadratic is in vertex form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y=a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=20 ALIGN=MIDDLE  ALT=\"a=20\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-39%2F40 ALIGN=MIDDLE  ALT=\"h=-39%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-881%2F80 ALIGN=MIDDLE  ALT=\"k=-881%2F80\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Remember (h,k) is the vertex and \"a\" is the stretch/compression factor.<BR>\n" );
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document.write( "  Check:<BR>\n" );
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document.write( "  Notice if we graph the original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=20x%5E2%2B39x%2B8 ALIGN=MIDDLE  ALT=\"y=20x%5E2%2B39x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C20x%5E2%2B39x%2B8%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C20x%5E2%2B39x%2B8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=20x%5E2%2B39x%2B8 ALIGN=MIDDLE  ALT=\"y=20x%5E2%2B39x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how the vertex is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-39%2F40 ALIGN=MIDDLE  ALT=\"-39%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-881%2F80 ALIGN=MIDDLE  ALT=\"-881%2F80\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
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document.write( "  Notice if we graph the final equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=20%28x%2B39%2F40%29%5E2-881%2F80 ALIGN=MIDDLE  ALT=\"y=20%28x%2B39%2F40%29%5E2-881%2F80\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C20%28x%2B39%2F40%29%5E2-881%2F80%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C20%28x%2B39%2F40%29%5E2-881%2F80%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=20%28x%2B39%2F40%29%5E2-881%2F80 ALIGN=MIDDLE  ALT=\"y=20%28x%2B39%2F40%29%5E2-881%2F80\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how the vertex is also (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-39%2F40 ALIGN=MIDDLE  ALT=\"-39%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-881%2F80 ALIGN=MIDDLE  ALT=\"-881%2F80\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
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document.write( "  So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.<BR>\n" );
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