Determine all pairs of real numbers $a$ and $b$, $b > 0$, such that the solutions to the two equations $$x^2 + ax + a = b \qquad \text{and} \qquad x^2 + ax + a = -b$$are four consecutive integers.
Source: SAMO 2016 Q2
Tags: quadratic equation, algebra
Determine all pairs of real numbers $a$ and $b$, $b > 0$, such that the solutions to the two equations $$x^2 + ax + a = b \qquad \text{and} \qquad x^2 + ax + a = -b$$are four consecutive integers.