document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=154113>Question 154113</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What three techniques can be used to solve a quadratic equation? Demonstrate these techniques on the equation \"12x^2 - 10x - 42 = 0\". </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #113542 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=113542\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> What three techniques can be used to solve a quadratic equation? Demonstrate these techniques on the equation \"12x^2 - 10x - 42 = 0\".<BR>\n" );
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document.write( "3 methods are:<BR>\n" );
document.write( "Factoring<BR>\n" );
document.write( "Completing the square<BR>\n" );
document.write( "Quadratic formula<BR>\n" );
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document.write( "12x^2 - 10x - 42 = 0<BR>\n" );
document.write( "Divide by 2.  Not necessary, but it makes it simpler to factor.<BR>\n" );
document.write( "6x^2 - 5x - 21 = 0<BR>\n" );
document.write( "Factoring is a trial and error process. If we use the quadratic equation to solve, we will know the factors.<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=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic 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;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B-5x%2B-21+=+0 ALIGN=MIDDLE  ALT=\"6x%5E2%2B-5x%2B-21+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-5%29%5E2-4%2A6%2A-21=529 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-5%29%5E2-4%2A6%2A-21=529\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=529 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--5%2B-sqrt%28+529+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--5%2B-sqrt%28+529+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+529+%29%29%2F2%5C6+=+2.33333333333333 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+529+%29%29%2F2%5C6+=+2.33333333333333\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-5%29-sqrt%28+529+%29%29%2F2%5C6+=+-1.5 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-5%29-sqrt%28+529+%29%29%2F2%5C6+=+-1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B-5x%2B-21 ALIGN=MIDDLE  ALT=\"6x%5E2%2B-5x%2B-21\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B-5x%2B-21+=+%28x-2.33333333333333%29%2A%28x--1.5%29 ALIGN=MIDDLE  ALT=\"6x%5E2%2B-5x%2B-21+=+%28x-2.33333333333333%29%2A%28x--1.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 2.33333333333333, -1.5.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+6%2Ax%5E2%2B-5%2Ax%2B-21+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+6%2Ax%5E2%2B-5%2Ax%2B-21+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "The factors shown by the on-site solver are not exactly right, it divides so the the coefficient of the x^2 is one.<BR>\n" );
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document.write( "Factoring:<BR>\n" );
document.write( "Using the results from above, we can multiply by 3 and 2 and get the factors.<BR>\n" );
document.write( "(3x - 7)*(2x + 3) = 6x^2 - 5x - 21<BR>\n" );
document.write( "So, 3x - 7 = 0<BR>\n" );
document.write( "x = 7/3<BR>\n" );
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document.write( "2x + 3 = 0<BR>\n" );
document.write( "x = -3/2<BR>\n" );
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document.write( "3rd method, Completing the square:<BR>\n" );
document.write( "12x^2 - 10x - 42 = 0<BR>\n" );
document.write( "Divide by 12<BR>\n" );
document.write( "x^2 - (5x/6) - 7/2 = 0<BR>\n" );
document.write( "The x term's coeff, -5/6, is 2 times the sqrt of the numeric term, so the NM, the last term will be (-5/12)^2, or 25/144.<BR>\n" );
document.write( "x^2 - (5x/6) - 7/2 = 0<BR>\n" );
document.write( "x^2 - (5x/6) = 7/2<BR>\n" );
document.write( "x^2 - (5x/6) + 25/144 = 7/2 + 25/144<BR>\n" );
document.write( "(x - 5/12)^2 = 7/2 + 25/144 = 504/144 + 25/144 = 529/144 = (23/12)^2<BR>\n" );
document.write( "(x - 5/12)^2 = (23/12)^2<BR>\n" );
document.write( "Take sqrt of both sides:<BR>\n" );
document.write( "x - 5/12 = 23/12 or -23/12<BR>\n" );
document.write( "x = 28/12 and x = -18/12<BR>\n" );
document.write( "x = 7/3 and -3/2<BR>\n" );
document.write( "Same answers, but a lot more work.  That's why we do the completion of the square ONE TIME with literal terms, ax^2 + bx + c = 0, to find the quadratic equation, then NEVER use completion of squares again.<BR>\n" );
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document.write( "BTW, a 4th method is to use Excel, or manual methods, to graph the function and find where it crosses the x-axis.\r<BR>\n" );
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