Question 9
Let $p, q, r$ be positive real numbers, not all equal, such that some two of the equations
$$
p x^{2}+2 q x+r=0, \quad q x^{2}+2 r x+p=0, \quad r x^{2}+2 p x+q=0
$$
have a common root, say $\alpha$. Prove that
(a) $\alpha$ is real and negative; and
(b) the third equation has non-real roots.
$$
p x^{2}+2 q x+r=0, \quad q x^{2}+2 r x+p=0, \quad r x^{2}+2 p x+q=0
$$
have a common root, say $\alpha$. Prove that
(a) $\alpha$ is real and negative; and
(b) the third equation has non-real roots.