Solving octic equation using quadratic formula

Question

According to the wikipedia article on octic equations, octic equations of the form $ax^8 \pm bx^4 \pm c = 0$ can be solved using the quadratic formula. How might one actually do this?

Answer 1

Replacing $w=x^4$ it's familiar:

$$w = \frac{-b \pm \sqrt{b^2 \pm 4ac}}{2a}.$$

Then extract the eight roots by taking $v_1 = \sqrt{w}, v_2 = -\sqrt{w}$, and then $x=\pm \sqrt{v_1}$ and $x = \pm \sqrt{v_2}$, for each of the two values of $w$.