If for some , not all have same sign, one of the roots of the equation is also a root…

MATHEMATICS · Subjective · AoPS · Mathematics · Quadratic Equations - Part 1 · Quadratic Equations - Part 1

Problem

If for some

p, q, r \in R

, not all have same sign, one of the roots of the equation

(p^2 + q^2)x^2 - 2q(p + r)x + q^2 + r^2 = 0

is also a root of the equation

x^2 + 2x - 8 = 0

, then

\frac{q^2 + r^2}{p^2}

is equal to \_\_\_\_.