document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=86802>Question 86802</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve the equation by completeing the square\r<BR>\n" );
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document.write( "a2(squared)-12a+27=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #62804 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=62804\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> note: I'm going to use \"x\" instead of \"a\" as my variable\r<BR>\n" );
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document.write( "We can convert any quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > to standard vertex form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > by this procedure:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%29%2Ax%5E2-%2812%29%2Ax%2B27 ALIGN=MIDDLE  ALT=\"y=%281%29%2Ax%5E2-%2812%29%2Ax%2B27\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given quadratic\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-27=%281%29%2Ax%5E2-%2812%29%2Ax ALIGN=MIDDLE  ALT=\"y-27=%281%29%2Ax%5E2-%2812%29%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27 ALIGN=MIDDLE  ALT=\"27\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-27=%281%29%2A%28x%5E2-%2812%29%2Ax%29 ALIGN=MIDDLE  ALT=\"y-27=%281%29%2A%28x%5E2-%2812%29%2Ax%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=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now to complete the square on the right side we must take half of the x coefficient (in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > its b) and square it  (i.e. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28b%2F2%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=y-27=%281%29%2A%28x%5E2-%2812%29%2Ax%2B36%29 ALIGN=MIDDLE  ALT=\"y-27=%281%29%2A%28x%5E2-%2812%29%2Ax%2B36%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take half of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12 ALIGN=MIDDLE  ALT=\"-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > and square it (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-12%29%281%2F2%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28%28-12%29%281%2F2%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >). Add the result(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36 ALIGN=MIDDLE  ALT=\"36\"   DISABLED_style= \"max-width:90%; height:auto;\" >) just inside the parenthesis.\r<BR>\n" );
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document.write( "This completes the square on the right side. So it goes from<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2A%28x%5E2-%2812%29%2Ax%2B36%29 ALIGN=MIDDLE  ALT=\"%281%29%2A%28x%5E2-%2812%29%2Ax%2B36%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2A%28x-6%29%5E2 ALIGN=MIDDLE  ALT=\"%281%29%2A%28x-6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is a perfect square\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-27=%281%29%2A%28x-6%29%5E2 ALIGN=MIDDLE  ALT=\"y-27=%281%29%2A%28x-6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the right side into a perfect square\r<BR>\n" );
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document.write( "Since we added <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36 ALIGN=MIDDLE  ALT=\"36\"   DISABLED_style= \"max-width:90%; height:auto;\" > inside the parenthesis, we really added <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2836%29 ALIGN=MIDDLE  ALT=\"%281%29%2836%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the entire right side (just distribute the leading coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you'll see it). So we must add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2836%29 ALIGN=MIDDLE  ALT=\"%281%29%2836%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the other side to balance the equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-27%2B%281%29%2A%2836%29=%281%29%2A%28x-6%29%5E2 ALIGN=MIDDLE  ALT=\"y-27%2B%281%29%2A%2836%29=%281%29%2A%28x-6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2836%29 ALIGN=MIDDLE  ALT=\"%281%29%2836%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the other side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-27%2B36=%281%29%2A%28x-6%29%5E2 ALIGN=MIDDLE  ALT=\"y-27%2B36=%281%29%2A%28x-6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-27%2B36=%281%29%2A%28x-6%29%5E2 ALIGN=MIDDLE  ALT=\"y-27%2B36=%281%29%2A%28x-6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B9=%281%29%2A%28x-6%29%5E2 ALIGN=MIDDLE  ALT=\"y%2B9=%281%29%2A%28x-6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the left side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B9=%281%29%2A%28x-6%29%5E2 ALIGN=MIDDLE  ALT=\"y%2B9=%281%29%2A%28x-6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce any fractions left side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%29%28x-6%29%5E2-9 ALIGN=MIDDLE  ALT=\"y=%281%29%28x-6%29%5E2-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides\r<BR>\n" );
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document.write( "So the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%29%2Ax%5E2-%2812%29%2Ax%2B27 ALIGN=MIDDLE  ALT=\"y=%281%29%2Ax%5E2-%2812%29%2Ax%2B27\"   DISABLED_style= \"max-width:90%; height:auto;\" > is completed to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%29%2A%28x-6%29%5E2-9 ALIGN=MIDDLE  ALT=\"y=%281%29%2A%28x-6%29%5E2-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is now in vertex form (which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"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 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the stretch/compression factor), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=6 ALIGN=MIDDLE  ALT=\"h=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >(the x-coordinate of the vertex), and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-9 ALIGN=MIDDLE  ALT=\"k=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the y coordinate of the vertex. So this means the vertex is <font size=6> (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9 ALIGN=MIDDLE  ALT=\"-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >)</font>. Also, since the axis of symmetry is the vertical line through the vertex, the axis of symmetry is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > (it is equal to the x-coordinate of the vertex).\r<BR>\n" );
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document.write( "Here are the graphs of original quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%29%2Ax%5E2-%2812%29%2Ax%2B27 ALIGN=MIDDLE  ALT=\"y=%281%29%2Ax%5E2-%2812%29%2Ax%2B27\"   DISABLED_style= \"max-width:90%; height:auto;\" > and our answer in vertex form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%29%28x-6%29%5E2-9 ALIGN=MIDDLE  ALT=\"y=%281%29%28x-6%29%5E2-9\"   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=%0D%0Adrawing%28+500%2C+500%2C+-16%2C+16%2C+-19%2C+19%2C%0D%0A++graph%28+500%2C+500%2C+-16%2C+16%2C+-19%2C+19%2C+1%2Ax%5E2%2B-12%2Ax%2B27%29%2C%0D%0A++red%28+locate%28+6%2C+-9%2C+%22%286%2C+-9%29%22+%29%2C+circle%28+6%2C+-9%2C+0.213333333333333+%29+%29%2C%0D%0A++graph%28+500%2C+500%2C+-16%2C+16%2C+-19%2C+19%2C+1000000%28x-6%29%29+%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%29%2Ax%5E2-%2812%29%2Ax%2B27 ALIGN=MIDDLE  ALT=\"y=%281%29%2Ax%5E2-%2812%29%2Ax%2B27\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the vertex <font size=6> (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9 ALIGN=MIDDLE  ALT=\"-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >)</font> and the axis of symmetry <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > (it is the vertical line through the vertex)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28+500%2C+500%2C+-16%2C+16%2C+-19%2C+19%2C%0D%0A++graph%28+500%2C+500%2C+-16%2C+16%2C+-19%2C+19%2C+1%28x-6%29%5E2%2B-9%29%2C%0D%0A++red%28+locate%28+6%2C+-9%2C+%22%286%2C+-9%29%22+%29%2C+circle%28+6%2C+-9%2C+0.213333333333333+%29+%29%2C%0D%0A++graph%28+500%2C+500%2C+-16%2C+16%2C+-19%2C+19%2C+1000000%28x-6%29%29+%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%29%28x-6%29%5E2-9 ALIGN=MIDDLE  ALT=\"y=%281%29%28x-6%29%5E2-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the vertex <font size=6> (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9 ALIGN=MIDDLE  ALT=\"-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >)</font> and the axis of symmetry <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > (it is the vertical line through the vertex)\r<BR>\n" );
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document.write( "Notice the two graphs are equivalent; this verifies our answer.\r<BR>\n" );
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document.write( "Now to solve for x, we simply need to isolate x: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=%281%29%2A%28x-6%29%5E2-9 ALIGN=MIDDLE  ALT=\"0=%281%29%2A%28x-6%29%5E2-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set y equal to zero to solve for x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2B9=%281%29%2A%28x-6%29%5E2 ALIGN=MIDDLE  ALT=\"0%2B9=%281%29%2A%28x-6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   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=0%2B-sqrt%289%29=x-6 ALIGN=MIDDLE  ALT=\"0%2B-sqrt%289%29=x-6\"   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/rendertex.cgi?\LARGE \pm3=x-6\"> Take the square root\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 6\pm3=x\"> Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides\r<BR>\n" );
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document.write( "So it breaks down to this\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6-3 ALIGN=MIDDLE  ALT=\"x=6-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6%2B3 ALIGN=MIDDLE  ALT=\"x=6%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our solution is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9 ALIGN=MIDDLE  ALT=\"x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice if you look back at the graph, you will see the roots <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9 ALIGN=MIDDLE  ALT=\"x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This verifies our answer </SPAN>\n" );
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