Find the factor of the given equation.

Question

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.

Answer 1

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 $k=0,1,2$.