document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=607617>Question 607617</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you solve 5x squared plus 16x minus 84 equals 0 using the quadratic equation\r<BR>\n" );
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document.write( "How do you solve 4x squared minus 12x plus 9 equals 0 using the quadratic equation </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #382854 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=382854\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are other ways to solve a quadratic equation, but using the quadratic formula always works if you are careful in doing your calculations.<BR>\n" );
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document.write( "SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA:<BR>\n" );
document.write( "A quadratic equation is an equation of the form<BR>\n" );
document.write( "<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;\" > where the coefficients b, and c are any number and a is not zero.<BR>\n" );
document.write( "Its solutions are given by the quadratic formula:<BR>\n" );
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;\" ><BR>\n" );
document.write( "To solve a quadratic equation by using the quadratic formula, you need to determine the values for a, b, and c in the equation you are solving. You then plug those values into the formula, and calculate very carefully.<BR>\n" );
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document.write( "FOR <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2%2B16x-84=0 ALIGN=MIDDLE  ALT=\"5x%5E2%2B16x-84=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, a=5, b=16, and c=-84,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-16+%2B-+sqrt%28+16%5E2-4%2A5%2A%28-84%29%29%29%2F%282%2A5%29=%28-16+%2B-+sqrt%28256%2B1680%29%29%2F10=%28-16+%2B-+sqrt%281936%29%29%2F10=%28-16+%2B-+44%29%2F10 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The answers are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-16%2B44%29%2F10=28%2F10=highlight%2814%2F5%29 ALIGN=MIDDLE  ALT=\"x=%28-16%2B44%29%2F10=28%2F10=highlight%2814%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-16-44%29%2F10=-60%2F10=highlight%28-6%29 ALIGN=MIDDLE  ALT=\"x=%28-16-44%29%2F10=-60%2F10=highlight%28-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "FOR <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-12x%2B9=0 ALIGN=MIDDLE  ALT=\"4x%5E2-12x%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, a=4, b=-12, and c=9,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-%28-12%29+%2B-+sqrt%28%28-12%29%5E2-4%2A4%2A9%29%29%2F%282%2A4%29=%2812+%2B-+sqrt%28144-144%29%29%2F8=%28-16+%2B-+sqrt%280%29%29%2F8=%2812+%2B-+0%29%2F8=12%2F8=3%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The one and only answer is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=3%2F2%29 ALIGN=MIDDLE  ALT=\"highlight%28x=3%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "THE OTHER WAYS TO SOLVE THOSE EQUATIONS:<BR>\n" );
document.write( "Factoring, with some work, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2%2B16x-84=%285x-14%29%28x%2B6%29 ALIGN=MIDDLE  ALT=\"5x%5E2%2B16x-84=%285x-14%29%28x%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2%2B16x-84=0 ALIGN=MIDDLE  ALT=\"5x%5E2%2B16x-84=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > turns into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x-14%29%28x%2B6%29=0 ALIGN=MIDDLE  ALT=\"%285x-14%29%28x%2B6%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > with solutions when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x-14=0 ALIGN=MIDDLE  ALT=\"5x-14=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=14%2F5 ALIGN=MIDDLE  ALT=\"x=14%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B6=0 ALIGN=MIDDLE  ALT=\"x%2B6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <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;\" ><BR>\n" );
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document.write( "The factoring is much easier for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-12x%2B9=0 ALIGN=MIDDLE  ALT=\"4x%5E2-12x%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, because it is obvious that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-12x%2B9=%282x-3%29%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2-12x%2B9=%282x-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > . It's the square of a binomial, one of the special products you are taught about in polynomials.<BR>\n" );
document.write( "The only solution for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-3%29%5E2=0 ALIGN=MIDDLE  ALT=\"%282x-3%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the value for x that makes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3=0 ALIGN=MIDDLE  ALT=\"2x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3=0 ALIGN=MIDDLE  ALT=\"2x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x=3 ALIGN=MIDDLE  ALT=\"2x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3%2F2 ALIGN=MIDDLE  ALT=\"x=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "FACTORING <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2%2B16x-84 ALIGN=MIDDLE  ALT=\"5x%5E2%2B16x-84\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "If it can be factored, it will be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2%2B16x-84=%285x%2Bd%29%28x%2Bf%29=5x%5E2%2B%28d%2B5f%29x%2Bdf%29 ALIGN=MIDDLE  ALT=\"5x%5E2%2B16x-84=%285x%2Bd%29%28x%2Bf%29=5x%5E2%2B%28d%2B5f%29x%2Bdf%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so we know that:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df=-84 ALIGN=MIDDLE  ALT=\"df=-84\"   DISABLED_style= \"max-width:90%; height:auto;\" > so d and f have different signs, and<BR>\n" );
document.write( "two factors (one positive and one negative) whose product is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Adf=-420=5%2A%28-84%29=%28d%29%2A%285f%29 ALIGN=MIDDLE  ALT=\"5%2Adf=-420=5%2A%28-84%29=%28d%29%2A%285f%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2B5f=16 ALIGN=MIDDLE  ALT=\"d%2B5f=16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There are many pairs of factors to 420 (we can worry about the signs later):<BR>\n" );
document.write( "420=(1)(420)=(2)(210)=(3)(140)=(4)(105)=(5)(84)=(6)(70)=(7)(60)=(10)(42)=(12)(35)=(14)(30)=(15)(28)=(20)(21),<BR>\n" );
document.write( "but the only pair that adds up to 16 (we consider sign now) is 30 and -14 (30-14=16).<BR>\n" );
document.write( "So, before collecting like term, the product could have been<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2%2B30x-14x-84 ALIGN=MIDDLE  ALT=\"5x%5E2%2B30x-14x-84\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that is easier to factor by grouping, as follows:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2%2B30x-14x-84=5x%28x%2B6%29-14%28x%2B6%29=%285x-14%29%28x%2B6%29 ALIGN=MIDDLE  ALT=\"5x%5E2%2B30x-14x-84=5x%28x%2B6%29-14%28x%2B6%29=%285x-14%29%28x%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );