
7. Find all real values of $a$ for which the equation $x^{4}-2 a x^{2}+x+a^{2}-a=0$ has all its roots real.


Solution: Let us consider $x^{4}-2 a x^{2}+x+a^{2}-a=0$ as a quadratic equation in $a$. We see that thee roots are

$$
a=x^{2}+x, \quad a=x^{2}-x+1
$$

Thus we get a factorisation

$$
\left(a-x^{2}-x\right)\left(a-x^{2}+x-1\right)=0
$$

It follows that $x^{2}+x=a$ or $x^{2}-x+1=a$. Solving these we get

$$
x=\frac{-1 \pm \sqrt{1+4 a}}{2}, \quad \text { or } \quad x=\frac{-1 \pm \sqrt{4 a-3}}{2}
$$

Thus all the four roots are real if and only if $a \geq 3 / 4$.

