Solving $2 + \frac 1{\cos^2 x} + \frac 5{\cos x} = 0$

Question

Simple but confusing trig question:

$$2 + \frac 1{\cos^2 x} + \frac 5{\cos x} = 0$$

I could not find anything in my textbook similar to this....

Answer 1

Hint: by multiplying both sides by $\cos^2 x$, this is the same equation as $$ 2\cos^2x + 1 + 5\cos x = 0, $$ which is a quadratic equation in $\cos x$.

Answer 2

$2y^2+ 5y+ 1= 0$

By the quadratic formula

$y=\dfrac{-5\pm \sqrt{25-8}}{4}$