document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1095189>Question 1095189</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you please solve this equation using Cardano's method:<BR>\n" );
document.write( "x^3+24x-25=0  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #709833 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=709833\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The general form for a cubic equation is<BR>\n" );
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document.write( "ax^3 + bx^2 + cx + d = 0<BR>\n" );
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document.write( "the problem's cubic equation is<BR>\n" );
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document.write( "x^3 + 24x - 25 = 0<BR>\n" );
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document.write( "this is a depressed cubic equation, the general form is x^3 + Ax = B<BR>\n" );
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document.write( "therefore we have<BR>\n" );
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document.write( "x^3 + 24x = 25<BR>\n" );
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document.write( "we determine s and t so that<BR>\n" );
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document.write( "1) 3st = A<BR>\n" );
document.write( "2) s^3 - t^3 = B<BR>\n" );
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document.write( "Cardano's method tells us that x = s - t will be one of the roots of the cubic equation<BR>\n" );
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document.write( "Solving the equation 1 for s and substituting into equation 2, we get<BR>\n" );
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document.write( "(A/3t)^3 - t^3 = B<BR>\n" );
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document.write( "simplifying this we get<BR>\n" );
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document.write( "t^6 + Bt^3 - (A^3/27) = 0<BR>\n" );
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document.write( "let u = t^3, then we have a quadratic equation<BR>\n" );
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document.write( "u^2 + Bu - (A^3/27) = 0<BR>\n" );
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document.write( "referring to our depressed cubic equation, we need s and t to satisfy<BR>\n" );
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document.write( "3) 3st = 24<BR>\n" );
document.write( "4) s^3 - t^3 = 25<BR>\n" );
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document.write( "u^2 +25u -(24^3/27) = 0<BR>\n" );
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document.write( "using quadratic formula, we have<BR>\n" );
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document.write( "u = (-25 + square root(25^2 - 4 * 1 * (-512))) / (2*1) = 13.3505<BR>\n" );
document.write( "u = (-25 - square root(25^2 - 4 * 1 * (-512))) / (2*1) = -38.3505<BR>\n" );
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document.write( "we reject the negative solution for u, therefore<BR>\n" );
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document.write( "t^3 = 13.3505<BR>\n" );
document.write( "s^3 = t^3 + 25 = 13.3505 + 25 = 38.3508<BR>\n" );
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document.write( "x = (38.3505)^(1/3) - (13.3505)^(1/3) = 1.000000955 approximately 1<BR>\n" );
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document.write( "therefore one solution is x=1, the other two solutions are complex<BR>\n" );
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document.write( "(x-1)(x^2+x+25) = 0 = x^3 + 24x - 25<BR>\n" );
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document.write( "we see that the two solutions to x^2 + x + 25 = 0 (using quadratic formula) are<BR>\n" );
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document.write( "x = (1/2)(-1-3isquare root(11)<BR>\n" );
document.write( "x = (1/2)(-1+3isquare root(11)<BR>\n" );
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