document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=77570>Question 77570</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write each equation in the form y = a(x-h)2 + k:\r<BR>\n" );
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document.write( "y = 2x2 + 20x + 50 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #55634 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=55634\">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=2+x%5E2%2B20+x%2B50 ALIGN=MIDDLE  ALT=\"y=2+x%5E2%2B20+x%2B50\"   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-50=2+x%5E2%2B20+x ALIGN=MIDDLE  ALT=\"y-50=2+x%5E2%2B20+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50 ALIGN=MIDDLE  ALT=\"50\"   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-50=2%28x%5E2%2B10x%29 ALIGN=MIDDLE  ALT=\"y-50=2%28x%5E2%2B10x%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=2 ALIGN=MIDDLE  ALT=\"2\"   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=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2810%29=5 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2810%29=5\"   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=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25 ALIGN=MIDDLE  ALT=\"25\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%29%5E2=%285%29%285%29=25 ALIGN=MIDDLE  ALT=\"%285%29%5E2=%285%29%285%29=25\"   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-50=2%28x%5E2%2B10x%2B25-25%29 ALIGN=MIDDLE  ALT=\"y-50=2%28x%5E2%2B10x%2B25-25%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=25 ALIGN=MIDDLE  ALT=\"25\"   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-50=2%28%28x%2B5%29%5E2-25%29 ALIGN=MIDDLE  ALT=\"y-50=2%28%28x%2B5%29%5E2-25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B10x%2B25 ALIGN=MIDDLE  ALT=\"x%5E2%2B10x%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B5%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B5%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-50=2%28x%2B5%29%5E2-2%2825%29 ALIGN=MIDDLE  ALT=\"y-50=2%28x%2B5%29%5E2-2%2825%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-50=2%28x%2B5%29%5E2-50 ALIGN=MIDDLE  ALT=\"y-50=2%28x%2B5%29%5E2-50\"   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=2%28x%2B5%29%5E2-50%2B50 ALIGN=MIDDLE  ALT=\"y=2%28x%2B5%29%5E2-50%2B50\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50 ALIGN=MIDDLE  ALT=\"50\"   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=2%28x%2B5%29%5E2%2B0 ALIGN=MIDDLE  ALT=\"y=2%28x%2B5%29%5E2%2B0\"   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=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-5 ALIGN=MIDDLE  ALT=\"h=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=0 ALIGN=MIDDLE  ALT=\"k=0\"   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=2x%5E2%2B20x%2B50 ALIGN=MIDDLE  ALT=\"y=2x%5E2%2B20x%2B50\"   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%2C2x%5E2%2B20x%2B50%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2x%5E2%2B20x%2B50%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%5E2%2B20x%2B50 ALIGN=MIDDLE  ALT=\"y=2x%5E2%2B20x%2B50\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how the vertex is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   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=2%28x%2B5%29%5E2%2B0 ALIGN=MIDDLE  ALT=\"y=2%28x%2B5%29%5E2%2B0\"   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%2C2%28x%2B5%29%5E2%2B0%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2%28x%2B5%29%5E2%2B0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%28x%2B5%29%5E2%2B0 ALIGN=MIDDLE  ALT=\"y=2%28x%2B5%29%5E2%2B0\"   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=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   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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document.write( "So the equation is now in vertex form where a=2, h=-5, k=0 and the vertex, which is (h,k), is:\r<BR>\n" );
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document.write( "vertex:  (-5,0) </SPAN>\n" );
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