Question

Solve the following quadratic equation, $2(x^2+\frac{1}{x^2})-9(x-\frac{1}{x})+14=0$.

Answer 1

Given equation is $2(x^2+\frac{1}{x^2})-9(x-\frac{1}{x})+14$,

To solve this consider $x-\frac{1}{x}=t$

$⟹x^2+\frac{1}{x^2}-2=t^2$,

Now, the equation transforms to

$2(t^2+2)-9t+14=0$

$⟹2t^2-9t+18=0$

$⟹t=\frac{3\pm i\sqrt{7}}{2}$

$⟹x-\frac{1}{x}=\frac{3\pm i\sqrt{7}}{2}$

$⟹x^2-\frac{3\pm i\sqrt{7}}{2}x-1=0$

$⟹x=\frac{\left(\frac{3\pm i\sqrt{7}}{2}\right)\pm \sqrt{\frac{1}{2}\pm \frac{3i\sqrt{7}}{2}+4}}{2}$