document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=84864>Question 84864</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve by completing the square.<BR>\n" );
document.write( "	4x^2 + 2x – 3 = 0<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #61161 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=61161\">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=4+x%5E2%2B2+x-3 ALIGN=MIDDLE  ALT=\"y=4+x%5E2%2B2+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=4+x%5E2%2B2+x ALIGN=MIDDLE  ALT=\"y%2B3=4+x%5E2%2B2+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=4%28x%5E2%2B%281%2F2%29x%29 ALIGN=MIDDLE  ALT=\"y%2B3=4%28x%5E2%2B%281%2F2%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=4 ALIGN=MIDDLE  ALT=\"4\"   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=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%281%2F2%29=1%2F4 ALIGN=MIDDLE  ALT=\"%281%2F2%29%281%2F2%29=1%2F4\"   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=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F16 ALIGN=MIDDLE  ALT=\"1%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F4%29%5E2=%281%2F4%29%281%2F4%29=1%2F16 ALIGN=MIDDLE  ALT=\"%281%2F4%29%5E2=%281%2F4%29%281%2F4%29=1%2F16\"   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=4%28x%5E2%2B%281%2F2%29x%2B1%2F16-1%2F16%29 ALIGN=MIDDLE  ALT=\"y%2B3=4%28x%5E2%2B%281%2F2%29x%2B1%2F16-1%2F16%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=1%2F16 ALIGN=MIDDLE  ALT=\"1%2F16\"   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=4%28%28x%2B1%2F4%29%5E2-1%2F16%29 ALIGN=MIDDLE  ALT=\"y%2B3=4%28%28x%2B1%2F4%29%5E2-1%2F16%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%281%2F2%29x%2B1%2F16 ALIGN=MIDDLE  ALT=\"x%5E2%2B%281%2F2%29x%2B1%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%2F4%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B1%2F4%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=4%28x%2B1%2F4%29%5E2-4%281%2F16%29 ALIGN=MIDDLE  ALT=\"y%2B3=4%28x%2B1%2F4%29%5E2-4%281%2F16%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=4%28x%2B1%2F4%29%5E2-1%2F4 ALIGN=MIDDLE  ALT=\"y%2B3=4%28x%2B1%2F4%29%5E2-1%2F4\"   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=4%28x%2B1%2F4%29%5E2-1%2F4-3 ALIGN=MIDDLE  ALT=\"y=4%28x%2B1%2F4%29%5E2-1%2F4-3\"   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=4%28x%2B1%2F4%29%5E2-13%2F4 ALIGN=MIDDLE  ALT=\"y=4%28x%2B1%2F4%29%5E2-13%2F4\"   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=4 ALIGN=MIDDLE  ALT=\"a=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-1%2F4 ALIGN=MIDDLE  ALT=\"h=-1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-13%2F4 ALIGN=MIDDLE  ALT=\"k=-13%2F4\"   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=4x%5E2%2B2x-3 ALIGN=MIDDLE  ALT=\"y=4x%5E2%2B2x-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%2C4x%5E2%2B2x-3%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C4x%5E2%2B2x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4x%5E2%2B2x-3 ALIGN=MIDDLE  ALT=\"y=4x%5E2%2B2x-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=-1%2F4 ALIGN=MIDDLE  ALT=\"-1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13%2F4 ALIGN=MIDDLE  ALT=\"-13%2F4\"   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=4%28x%2B1%2F4%29%5E2-13%2F4 ALIGN=MIDDLE  ALT=\"y=4%28x%2B1%2F4%29%5E2-13%2F4\"   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%2C4%28x%2B1%2F4%29%5E2-13%2F4%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C4%28x%2B1%2F4%29%5E2-13%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4%28x%2B1%2F4%29%5E2-13%2F4 ALIGN=MIDDLE  ALT=\"y=4%28x%2B1%2F4%29%5E2-13%2F4\"   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=-1%2F4 ALIGN=MIDDLE  ALT=\"-1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13%2F4 ALIGN=MIDDLE  ALT=\"-13%2F4\"   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( "Since we have \r<BR>\n" );
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document.write( "Set y=0 and solve for x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%2F4=4%28x%2B1%2F4%29%5E2 ALIGN=MIDDLE  ALT=\"13%2F4=4%28x%2B1%2F4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%2F4 ALIGN=MIDDLE  ALT=\"13%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2813%2F4%29%281%2F4%29=%28x%2B1%2F4%29%5E2 ALIGN=MIDDLE  ALT=\"%2813%2F4%29%281%2F4%29=%28x%2B1%2F4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   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=0%2B-sqrt%2813%2F16%29=x%2B1%2F4 ALIGN=MIDDLE  ALT=\"0%2B-sqrt%2813%2F16%29=x%2B1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2B-sqrt%2813%29%2F4-1%2F4=x ALIGN=MIDDLE  ALT=\"0%2B-sqrt%2813%29%2F4-1%2F4=x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides\r<BR>\n" );
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document.write( "So our answer is\r<BR>\n" );
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document.write( "which breaks down to \r<BR>\n" );
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