document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=276593>Question 276593</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I find this quite difficult, though I am gaining some level of understanding.  I am unsure how to solve for the following 2 quadratic equations  Please help<BR>\n" );
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document.write( "1. x2 – 2x – 13 = 0<BR>\n" );
document.write( "2. 2x2 – 3x – 5 =0 \r<BR>\n" );
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document.write( "These are the steps I need to take, just unsure how\r<BR>\n" );
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document.write( "a. move the constant term to the right side of the equation<BR>\n" );
document.write( "b. multiply each term in the equation by four times the cofficient of the x2 term<BR>\n" );
document.write( "c. square the coefficient of the original x term and add it to both sides of the equation<BR>\n" );
document.write( "d. take the square root of both sides<BR>\n" );
document.write( "e. set the left side of the equation equal to the positive square root of the number on the right side and solve for x<BR>\n" );
document.write( "f. set the left side of the equation equal to the negative square root of the number on the right side of the equation and solve for x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #201574 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=201574\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "What you have is (at least to me) a rather unorthodox method of a process called \"completing the square.\"  Despite the very unfamiliar process, I have proven to myself that it actually works in the general case.\r<BR>\n" );
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document.write( "Start with the general form of the quadratic equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ ax^2\ +\ bx\ +\ c\ =\ 0\">\r<BR>\n" );
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document.write( "Now, let's follow your steps:\r<BR>\n" );
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document.write( "\"a. move the constant term to the right side of the equation\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ ax^2\ +\ bx\ =\ -c\">\r<BR>\n" );
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document.write( "\"b. multiply each term in the equation by four times the cofficient of the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\"> term\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4a^2x^2\ +\ 4abx\ =\ -4ac\">\r<BR>\n" );
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document.write( "\"c. square the coefficient of the original x term and add it to both sides of the equation\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4a^2x^2\ +\ 4abx\ +\ b^2\ =\ b^2\ -\ 4ac\">\r<BR>\n" );
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document.write( "\"d. take the square root of both sides\"\r<BR>\n" );
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document.write( "First you need to factor the expression on the LHS of the equation.  Fortunately, this is a perfect square expression:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(2ax\ +\ b\right)^2\ =\ b^2\ -\ 4ac\">\r<BR>\n" );
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document.write( "(Verification of that last intermediate step is left as an exercise for the student)\r<BR>\n" );
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document.write( "Now we can take the square root of both sides, remembering to consider both the positive and negative roots -- such consideration typically indicated in the resulting RHS.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2ax\ +\ b\ =\ \pm\sqrt{b^2\ -\ 4ac}\">\r<BR>\n" );
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document.write( "By considering both positive and negative roots at this point, you can do steps e and f simultaneously:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{-b\ \pm\sqrt{b^2\ -\ 4ac}}{2a}\">\r<BR>\n" );
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document.write( "And we achieve a result that is identical to the quadratic formula which is the solution to the general quadratic.\r<BR>\n" );
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document.write( "Let's do one of your problems now that we are certain that the process will work for every problem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2\ -\ 2x\ -\ 13\ =\ 0\">\r<BR>\n" );
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document.write( "a:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2\ -\ 2x\ =\ 13\">\r<BR>\n" );
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document.write( "b:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4x^2\ -\ 8x\ =\ 52\">\r<BR>\n" );
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document.write( "c:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4x^2\ -\ 8x\ +\ 4\ =\ 56\">\r<BR>\n" );
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document.write( "d:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (2x\ -\ 2)^2\ =\ 56\">\r<BR>\n" );
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document.write( "d1: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2x\ -\ 2\ =\ \pm\sqrt{56}\">\r<BR>\n" );
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document.write( "e&f: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{2\ \pm\sqrt{56}}{2}\">\r<BR>\n" );
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document.write( "Then you can simplify a little by recognizing that 56 is 4 times 14, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 1\ \pm\sqrt{14}\">\r<BR>\n" );
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document.write( "Let's see if this answer makes any sense.  3 squared is 9 and 4 squared is 16, so 3.75 is a pretty good rough guess for square root 14.  1 plus 3.75 is 4.75 and 1 minus 3.75 is -2.75.  So look where the graph crosses the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-axis:\r<BR>\n" );
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document.write( "Pretty close, I'd say.  Let me know if you still need help.\r<BR>\n" );
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document.write( "John<BR>\n" );
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