document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=373661>Question 373661</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need to solve these quadratic equations using the method  that came from India.  I am confused. \r<BR>\n" );
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document.write( "The sample problem is:<BR>\n" );
document.write( "xsquared + 3x -10 = 0\r<BR>\n" );
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document.write( "The solution reads as follows, but I do not understand it.\r<BR>\n" );
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document.write( "x squared + 3x = 10<BR>\n" );
document.write( "4x squared + 12x = 40<BR>\n" );
document.write( "4x squared + 12x +9 = 40 + 9<BR>\n" );
document.write( "4x squared + 12x +9 = 49<BR>\n" );
document.write( "2x + 3 = +-7\r<BR>\n" );
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document.write( "2x + 3 = 7<BR>\n" );
document.write( "2x = 4<BR>\n" );
document.write( "x = 2 <BR>\n" );
document.write( "and<BR>\n" );
document.write( "2x + 3 = -7<BR>\n" );
document.write( "2x = -10<BR>\n" );
document.write( "x = -5\r<BR>\n" );
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document.write( "Is there anyway you can explain this method to me?  I do not understand where the x squared turned into the 4x squared and where the 12x went later in the problem.  Thank you in advance for your help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #265934 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=265934\">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( "This so-called \"Indian\" method is just a variation of completing the square.  I suspect you are being introduced to the method as a practical illustration of the fact that in mathematics, as well as most other fields of human endeavor, there is almost always more than one way to skin a cat.\r<BR>\n" );
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document.write( "In general, the Indian Method is as follows:\r<BR>\n" );
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document.write( "Given\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( "Step 1:  Add the opposite of the constant term 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 \ \ \ \ \ \ \ \ \ \ ax^2\ +\ bx\ =\ -c\">\r<BR>\n" );
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document.write( "Step 2:  Multiply both sides by 4 times the original lead coefficient:\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( "Step 3:  Square the original coefficient on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> term and add that to both sides:\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( "Step 4:  The trinomial in the LHS is now a perfect square, so factor it:\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( "Step 5:  Take the square root of both sides, remembering to consider both the positive and negative roots:\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( "Step 6:  Add the opposite of the constant term in the LHS to both sides and then divide both sides by the coefficient on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> term in the LHS\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( "Which you should recognize as the quadratic formula.\r<BR>\n" );
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document.write( "Given\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2\ +\ 3x\ -\ 10\ =\ 0\">\r<BR>\n" );
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document.write( "Note that \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ =\ 1\ \ \">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b\ =\ 3\ \ \">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE c\ =\ -10\">\r<BR>\n" );
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document.write( "Step 1:  Add the opposite of the constant term 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 \ \ \ \ \ \ \ \ \ \ x^2\ +\ 3x\ =\ 10\">\r<BR>\n" );
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document.write( "Step 2:  Multiply both sides by 4 times the original lead coefficient (<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 4\,\cdot\,1\ =\ 4\">):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4x^2\ +\ (4\,\cdot\,3)x\ =\ 4\,\cdot\,10\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4x^2\ +\ 12x\ =\ 40\">\r<BR>\n" );
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document.write( "Step 3:  Square the original coefficient on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> term and add that to both sides (<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\ =\ 3\ \rightarrow\ b^2\ =\ 9\">):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4x^2\ +\ 12x\ +\ 9\ =\ 9\ +\ 40\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4x^2\ +\ 12x\ +\ 9\ =\ 49\">\r<BR>\n" );
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document.write( "Step 4:  The trinomial in the LHS is now a perfect square, so factor it:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(2x\ +\ 3\right)^2\ =\ 49\">\r<BR>\n" );
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document.write( "Step 5:  Take the square root of both sides, remembering to consider both the positive and negative roots:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2x\ +\ 3\ =\ \pm\sqrt{49}\">\r<BR>\n" );
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document.write( "Step 6:  Add the opposite of the constant term in the LHS to both sides and then divide both sides by the coefficient on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> term in the LHS\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{-3\ \pm\ \sqrt{49}}{2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{-10}{2}\ =\ -5\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{4}{2}\ =\ 2\">\r<BR>\n" );
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document.write( "Check your work:\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \beta\"> are the roots of:\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( "then \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\ -\ \alpha)(x\ -\ \beta)\ =\ ax^2\ +\ bx\ +\ c\">\r<BR>\n" );
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document.write( "So use FOIL to multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\ -\ (-5))(x\ -\ 2)\ =\ (x\ +\ 5)(x\ -\ 2)\"> to verify that the product is indeed\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2\ +\ 3x\ -\ 10\ =\ 0\">\r<BR>\n" );
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document.write( "Verification left as an exercise for the student.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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