Common root of a cubic and a biquadratic equation

Question

The value of $a$ given that the cubic equation

$$x^3+2ax+2=0$$

and the biquadratic equation
$$x^4+2ax^2+1=0$$

have a common root.

I know how to use common root condition for two quadratic equations, But I don't know how to solve this…

Answer 1

Assume $r$ is the common root. Then,

$$r^3+2ar+2=0\tag1$$ $$r^4+2ar^2+1=0\tag2$$

Take (1)$\cdot$r-(2) to obtain $r=\frac12$ and then $a=-\frac{17}8$.