document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=608402>Question 608402</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The coefficient of the second degree term of a quadratic equation is 1. If the constant term of the equation is increased by 1, the roots of the resulting equation are equal; but if it is diminished by 3, one root of the resulting equation is double the other. Find the original quadratic equation and its roots. So far, I've came up with ax^2+bx+(c-3) and ax^2+bx+(c-1). I don't know if I should be using the standard formula, vertex formula, root formula, etc. Please help me? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #383177 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=383177\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You just have to read every sentence carefully to see what it is saying. We know that the leading coefficient is 1, so the polynomial is in the form\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2 + bx + c\">\r<BR>\n" );
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document.write( "According to the second sentence, the polynomial <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2 + bx + (c+1)\"> has equal roots. This means that the discriminant is zero, i.e.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b^2 - 4(c+1) = 0 \Rightarrow b^2 - 4c = 4\">\r<BR>\n" );
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document.write( "According to the third sentence, the polynomial <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2 + bx + (c-3)\"> has two roots, one of which is double the other. Suppose we find the roots:\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 - 4(c-3)}}{2}\">\r<BR>\n" );
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document.write( "Therefore if we let -b = 3k and sqrt(b^2 - 4(c-3)) = k for some k, we will obtain two roots, one of which is double the other (since 3k + k = 2(3k - k)). Hence,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -b = 3\sqrt{b^2 - 4(c-3)} = 3\sqrt{b^2 - 4c + 12}\">\r<BR>\n" );
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document.write( "We have a system of two equations but this is easily solvable since we know that b^2 - 4c = 4. Substitute this to obtain\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -b = 3\sqrt{(4) + 12} = 12 \Rightarrow b = -12\">\r<BR>\n" );
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document.write( "Solving for c, we obtain c = 35. Therefore the original quadratic is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2 - 12x + 35\"> and the roots are 5 and 7 (solved by factoring).\r<BR>\n" );
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document.write( "Update: I let -b = 3k and sqrt(b^2 - 4(c-3)) = k since that will produce two roots in the ratio 2:1. The roots of the quadratic now become (3k+k)/2 and (3k-k)/2, which are equal to 2k and k, in the ratio 2:1. </SPAN>\n" );
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