MathNet 2008-part-01 — 165 problems (EN–VI) Question 91

Let $a$ be a real number such that the quadratic equation $x^2 - x + a = 0$ has two real distinct roots $x_1$ and $x_2$. Prove that $|x_1^2 - x_2^2| = 1$ if and only if $|x_1^3 - x_2^3| = 1$.