document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=119027>Question 119027</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780073268620/algebrahomeworkh>U of Phoenix elementry and intermediate algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> I know its got to be simple, but I just don't understand quadratic equations.  12x^+5x-3=0    ^=squared </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #87140 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=87140\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2%2B5x-3=0 ALIGN=MIDDLE  ALT=\"12x%5E2%2B5x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This particular quadratic is factorable, meaning that the roots of the equation are rational numbers, although this one takes a bit of work to find the factors.\r<BR>\n" );
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document.write( "A quicker method would be to use the quadratic formula and work backwards to find the factors.  Here's the quadratic formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where a is the coefficient on the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term, b is the coefficient on the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > term, and c is the constant.\r<BR>\n" );
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document.write( "In this problem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-5+%2B-+sqrt%28+5%5E2-4%2A12%2A%28-3%29+%29%29%2F%282%2A12%29=%28-5%2B-sqrt%28169%29%29%2F24+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.  Now we are certain that the roots of the equation will be rational because 169 is a perfect square, namely <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%5E2 ALIGN=MIDDLE  ALT=\"13%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now our two answers will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=%28-5%2B13%29%2F24=8%2F24=1%2F3 ALIGN=MIDDLE  ALT=\"x%5B1%5D=%28-5%2B13%29%2F24=8%2F24=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=%28-5-13%29%2F24=-18%2F24=-%283%2F4%29 ALIGN=MIDDLE  ALT=\"x%5B2%5D=%28-5-13%29%2F24=-18%2F24=-%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F3 ALIGN=MIDDLE  ALT=\"x=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-1=0 ALIGN=MIDDLE  ALT=\"3x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-%283%2F4%29 ALIGN=MIDDLE  ALT=\"x=-%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B3=0 ALIGN=MIDDLE  ALT=\"4x%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  These are the two factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2%2B5x-3 ALIGN=MIDDLE  ALT=\"12x%5E2%2B5x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, as you will see if you use FOIL to multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x-1%29%284x%2B3%29 ALIGN=MIDDLE  ALT=\"%283x-1%29%284x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  If either one of those factors is equal to zero, the entire expression is equal to zero.\r<BR>\n" );
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document.write( "Perhaps it bothers you that there are two answers.  Take a look at a graph of the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=12x%5E2%2B5x-3 ALIGN=MIDDLE  ALT=\"f%28x%29=12x%5E2%2B5x-3\"   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=graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C12x%5E2%2B5x-3%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C12x%5E2%2B5x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "See where the graph crosses the x-axis at two points?  Each of these points is a zero of the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=12x%5E2%2B5x-3 ALIGN=MIDDLE  ALT=\"f%28x%29=12x%5E2%2B5x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or a root of the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2%2B5x-3=0 ALIGN=MIDDLE  ALT=\"12x%5E2%2B5x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Hope this helps,<BR>\n" );
document.write( "John </SPAN>\n" );
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