How to solve a double quadratic inequality

Question

I have a double quadratic inequality of the form $2x^4 + 2x^2 - 3 < 0 $. Had it been equation rather than inequality I would have probably set $x^2 = z$ and looked for the solution in terms of z and finally in terms of x. But how to deal with this inequality?

Answer 1

It's $$x^2<\frac{-1+\sqrt{7}}{2}$$ or $$-\sqrt{\frac{-1+\sqrt{7}}{2}}<x<\sqrt{\frac{-1+\sqrt{7}}{2}}.$$

Answer 2

You can complete the square: $$2\left(x^2+\frac{1}{2}\right)^2-\frac{7}{2}<0,$$ or $$\left(x^2+\frac{1}{2}\right)^2<\frac{7}{4}$$ Can you see how to solve this?