document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=119297>Question 119297</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is  the standard form for y=1/2x^2+4x-3 ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #87399 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=87399\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\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=%281%2F2%29+x%5E2%2B4+x-3 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29+x%5E2%2B4+x-3\"   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%2B3=%281%2F2%29+x%5E2%2B4+x ALIGN=MIDDLE  ALT=\"y%2B3=%281%2F2%29+x%5E2%2B4+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B3=%281%2F2%29%28x%5E2%2B8x%29 ALIGN=MIDDLE  ALT=\"y%2B3=%281%2F2%29%28x%5E2%2B8x%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=%281%2F2%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29\"   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=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%288%29=4 ALIGN=MIDDLE  ALT=\"%281%2F2%29%288%29=4\"   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=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%29%5E2=%284%29%284%29=16 ALIGN=MIDDLE  ALT=\"%284%29%5E2=%284%29%284%29=16\"   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%2B3=%281%2F2%29%28x%5E2%2B8x%2B16-16%29 ALIGN=MIDDLE  ALT=\"y%2B3=%281%2F2%29%28x%5E2%2B8x%2B16-16%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=16 ALIGN=MIDDLE  ALT=\"16\"   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%2B3=%281%2F2%29%28%28x%2B4%29%5E2-16%29 ALIGN=MIDDLE  ALT=\"y%2B3=%281%2F2%29%28%28x%2B4%29%5E2-16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B8x%2B16 ALIGN=MIDDLE  ALT=\"x%5E2%2B8x%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B4%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%2B3=%281%2F2%29%28x%2B4%29%5E2-%281%2F2%29%2816%29 ALIGN=MIDDLE  ALT=\"y%2B3=%281%2F2%29%28x%2B4%29%5E2-%281%2F2%29%2816%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%2B3=%281%2F2%29%28x%2B4%29%5E2-8 ALIGN=MIDDLE  ALT=\"y%2B3=%281%2F2%29%28x%2B4%29%5E2-8\"   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=%281%2F2%29%28x%2B4%29%5E2-8-3 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29%28x%2B4%29%5E2-8-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2B3 ALIGN=MIDDLE  ALT=\"%2B3\"   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=%281%2F2%29%28x%2B4%29%5E2-11 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29%28x%2B4%29%5E2-11\"   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=1%2F2 ALIGN=MIDDLE  ALT=\"a=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-4 ALIGN=MIDDLE  ALT=\"h=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-11 ALIGN=MIDDLE  ALT=\"k=-11\"   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=%281%2F2%29x%5E2%2B4x-3 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29x%5E2%2B4x-3\"   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%2C%281%2F2%29x%5E2%2B4x-3%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%281%2F2%29x%5E2%2B4x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29x%5E2%2B4x-3 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29x%5E2%2B4x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how the vertex is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-11 ALIGN=MIDDLE  ALT=\"-11\"   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=%281%2F2%29%28x%2B4%29%5E2-11 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29%28x%2B4%29%5E2-11\"   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%2C%281%2F2%29%28x%2B4%29%5E2-11%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%281%2F2%29%28x%2B4%29%5E2-11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29%28x%2B4%29%5E2-11 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29%28x%2B4%29%5E2-11\"   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=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-11 ALIGN=MIDDLE  ALT=\"-11\"   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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