document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=766811>Question 766811</A>: <I><SPAN STYLE=\"word-break: break-all;\"> c)(i) Write the quadratic expression x^2-34x+9 in completed square form\r<BR>\n" );
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document.write( "(ii) use the completed square form from part (c) (i) to solve the equation x^2-34x+9=0, leaving your answer in exact (surd) form\r<BR>\n" );
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document.write( "(iii) use the completed square form from part (c)(i)to write down the vertex of the parabola y=x^2-34x+9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #467231 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=467231\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> c)(i) Write the quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-34x%2B9 ALIGN=MIDDLE  ALT=\"x%5E2-34x%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > in completed square form\r<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=1+x%5E2-34+x%2B9 ALIGN=MIDDLE  ALT=\"y=1+x%5E2-34+x%2B9\"   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-9=1+x%5E2-34+x ALIGN=MIDDLE  ALT=\"y-9=1+x%5E2-34+x\"   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<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-9=1%28x%5E2-34x%29 ALIGN=MIDDLE  ALT=\"y-9=1%28x%5E2-34x%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" );
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document.write( "  Take half of the x coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-34 ALIGN=MIDDLE  ALT=\"-34\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-17 ALIGN=MIDDLE  ALT=\"-17\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28-34%29=-17 ALIGN=MIDDLE  ALT=\"%281%2F2%29%28-34%29=-17\"   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=-17 ALIGN=MIDDLE  ALT=\"-17\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=289 ALIGN=MIDDLE  ALT=\"289\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-17%29%5E2=%28-17%29%28-17%29=289 ALIGN=MIDDLE  ALT=\"%28-17%29%5E2=%28-17%29%28-17%29=289\"   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-9=1%28x%5E2-34x%2B289-289%29 ALIGN=MIDDLE  ALT=\"y-9=1%28x%5E2-34x%2B289-289%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=289 ALIGN=MIDDLE  ALT=\"289\"   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-9=1%28%28x-17%29%5E2-289%29 ALIGN=MIDDLE  ALT=\"y-9=1%28%28x-17%29%5E2-289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-34x%2B289 ALIGN=MIDDLE  ALT=\"x%5E2-34x%2B289\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-17%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-17%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-9=1%28x-17%29%5E2-1%28289%29 ALIGN=MIDDLE  ALT=\"y-9=1%28x-17%29%5E2-1%28289%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-9=1%28x-17%29%5E2-289 ALIGN=MIDDLE  ALT=\"y-9=1%28x-17%29%5E2-289\"   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=1%28x-17%29%5E2-289%2B9 ALIGN=MIDDLE  ALT=\"y=1%28x-17%29%5E2-289%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now 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 to isolate y<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%28x-17%29%5E2-280 ALIGN=MIDDLE  ALT=\"y=1%28x-17%29%5E2-280\"   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 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=17 ALIGN=MIDDLE  ALT=\"h=17\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-280 ALIGN=MIDDLE  ALT=\"k=-280\"   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=1x%5E2-34x%2B9 ALIGN=MIDDLE  ALT=\"y=1x%5E2-34x%2B9\"   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%2C1x%5E2-34x%2B9%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1x%5E2-34x%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1x%5E2-34x%2B9 ALIGN=MIDDLE  ALT=\"y=1x%5E2-34x%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how the vertex is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17 ALIGN=MIDDLE  ALT=\"17\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-280 ALIGN=MIDDLE  ALT=\"-280\"   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=1%28x-17%29%5E2-280 ALIGN=MIDDLE  ALT=\"y=1%28x-17%29%5E2-280\"   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%2C1%28x-17%29%5E2-280%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1%28x-17%29%5E2-280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%28x-17%29%5E2-280 ALIGN=MIDDLE  ALT=\"y=1%28x-17%29%5E2-280\"   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=17 ALIGN=MIDDLE  ALT=\"17\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-280 ALIGN=MIDDLE  ALT=\"-280\"   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( "(ii) use the completed square form from part (c) (i) to solve the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-34x%2B9=0 ALIGN=MIDDLE  ALT=\"x%5E2-34x%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, leaving your answer in exact (surd) form\r<BR>\n" );
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document.write( "(iii) use the completed square form from part (c)(i)to write down the vertex of the parabola <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2-34x%2B9 ALIGN=MIDDLE  ALT=\"y=x%5E2-34x%2B9\"   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=%28x-17%29%5E2-280+ ALIGN=MIDDLE  ALT=\"y=%28x-17%29%5E2-280+\"   DISABLED_style= \"max-width:90%; height:auto;\" >....compare to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y=%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you will see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=17 ALIGN=MIDDLE  ALT=\"h=17\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-280 ALIGN=MIDDLE  ALT=\"k=-280\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means\r<BR>\n" );
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document.write( "the vertex of the parabola is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17 ALIGN=MIDDLE  ALT=\"17\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-280 ALIGN=MIDDLE  ALT=\"-280\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "since the graph above is not good, I will make another one for you to see it is a parabola:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-50%2C+50%2C+-330%2C+50%2C%28x-17%29%5E2-280+%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-50%2C+50%2C+-330%2C+50%2C%28x-17%29%5E2-280+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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