document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=187247>Question 187247</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the zeroes of a quadratic function are -4 and 6, and its y-intercept is 12, determine the function in general form without using regression.<BR>\n" );
document.write( "My name is Chantal and I know that -4 and 6 are the roots and -12 is the last part of the equation in general form but I don't know what he means by regression. I'm not sure how to go about doing this equation. If you could help me it would be greatly appreciated. I missed school when we learnt this.This is not from a book.<BR>\n" );
document.write( "Chantal Laroche </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #140352 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=140352\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \"If the zeroes of a quadratic function are -4 and 6\" means that the solutions to the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <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;\" > (ie if you plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <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;\" > into <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;\" >, you will get zero)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <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;\" > Start with the given solutions\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B4=0 ALIGN=MIDDLE  ALT=\"x%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-6=0 ALIGN=MIDDLE  ALT=\"x-6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Get the terms to the left side (so the right side is zero)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%28x-6%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%28x-6%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the equations using the zero product property\r<BR>\n" );
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document.write( "Note: the zero product property states that if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2AB=0 ALIGN=MIDDLE  ALT=\"A%2AB=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > then either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=0 ALIGN=MIDDLE  ALT=\"A=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=0 ALIGN=MIDDLE  ALT=\"B=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or possibly both)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x%2B4%29%28x-6%29=0 ALIGN=MIDDLE  ALT=\"a%28x%2B4%29%28x-6%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Place an additional variable \"a\" onto the front of the equation. Note: this will be needed to satisfy the third condition.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x%5E2-6x%2B4x-24%29=0 ALIGN=MIDDLE  ALT=\"a%28x%5E2-6x%2B4x-24%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x%5E2-2x-24%29=0 ALIGN=MIDDLE  ALT=\"a%28x%5E2-2x-24%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "So the equation that has the zeros of -4 and 6 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x%5E2-2x-24%29 ALIGN=MIDDLE  ALT=\"y=a%28x%5E2-2x-24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now there's an additional condition: The y-intercept is 12\r<BR>\n" );
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document.write( "This means that when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12 ALIGN=MIDDLE  ALT=\"y=12\"   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=a%28x%5E2-2x-24%29 ALIGN=MIDDLE  ALT=\"y=a%28x%5E2-2x-24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the previous equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12=a%280%5E2-2%280%29-24%29 ALIGN=MIDDLE  ALT=\"12=a%280%5E2-2%280%29-24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12 ALIGN=MIDDLE  ALT=\"y=12\"   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=12=a%280-2%280%29-24%29 ALIGN=MIDDLE  ALT=\"12=a%280-2%280%29-24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 0 to get 0\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12=a%280-0-24%29 ALIGN=MIDDLE  ALT=\"12=a%280-0-24%29\"   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=12=a%28-24%29 ALIGN=MIDDLE  ALT=\"12=a%28-24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F%28-24%29=a ALIGN=MIDDLE  ALT=\"12%2F%28-24%29=a\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by -24 to isolate \"a\".\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2=a ALIGN=MIDDLE  ALT=\"-1%2F2=a\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "So the value of \"a\" is <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;\" >\r<BR>\n" );
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document.write( "This means that the equation then goes from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x%5E2-2x-24%29 ALIGN=MIDDLE  ALT=\"y=a%28x%5E2-2x-24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29%28x%5E2-2x-24%29 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29%28x%5E2-2x-24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now distribute to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29%28x%5E2%29-%28-1%2F2%292x-%28-1%2F2%2924 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29%28x%5E2%29-%28-1%2F2%292x-%28-1%2F2%2924\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and multiply: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29x%5E2%2Bx%2B12 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29x%5E2%2Bx%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the equation with the zeros -4 and 6 with the y-intercept of 12 is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29x%5E2%2Bx%2B12 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29x%5E2%2Bx%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here's the graph to visually verify our answer:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A+graph%28+500%2C+500%2C+-10%2C+10%2C+-5%2C+15%2C%28-1%2F2%29x%5E2%2Bx%2B12%29%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29x%5E2%2Bx%2B12 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29x%5E2%2Bx%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" > with x-intercepts (-4,0) and (6,0) with the y-intercept (0,12) </SPAN>\n" );
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