Find the factor of the equation $x^2-42.5 x^{\frac{2}{3}}-78.4=0$ ? I have tried it by substituting $x^{\frac {2}{3}}$ by $z$ and get a cubic equation $z^3-42.5z-78.4=0$ and tried to solve it by using Cardan's method but it was too lengthy. Please help me to solve it in any simplest way. Thanks in advance.
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The roots are messy. Accept it. – Saketh Malyala Jun 15 '17 at 06:35
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1There's no way out no matter how hard you wish. also, check with wolfram alpha – MaudPieTheRocktorate Jun 15 '17 at 06:37
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Cardano's method is relatively simple – Ahmed S. Attaalla Jun 15 '17 at 06:39
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Check this out https://math.stackexchange.com/questions/2318064/solving-depressed-cubic-extra-solutions – Ahmed S. Attaalla Jun 15 '17 at 06:41
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But for this case cardan's method takes very huge calculation. How can I solve it for only 2 marks. – Iamdark Jun 15 '17 at 06:43
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To make things cleaner, consider the equation to be $$ z^3-\frac{425}{10}z -\frac{784}{10}=z^3-\frac{85 }{2}z-\frac{392}{5}=0$$ From the discriminant, you know that there are three real roots. So, use the trigonometric method for solving the roots and get $$Q=-\frac{85}{6}\qquad R=\frac{196}{5} \qquad \theta=\cos ^{-1}\left(\frac{1176 \sqrt{\frac{6}{85}}}{425}\right)$$ and then the three roots given by $$z_k=\sqrt{\frac{170}{3}} \cos \left(\frac {2k \pi}3+\frac{1}{3} \cos ^{-1}\left(\frac{1176 \sqrt{\frac{6}{85}}}{425}\right)\right)$$ using $k=0,1,2$.
Claude Leibovici
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1@Iamdark. You are welcome ! I hope I did not make any mistake with these numbers; but the idea is this. – Claude Leibovici Jun 15 '17 at 08:28