p is a nonzero real root of the equation: ax²+ bx + c = 0 and, q is a nonzero real root of the equation: -ax²+ bx + c = 0.

Does the equation ax²/2 + bx + c = 0 always have a root between p and q?

If so, prove it. Otherwise, give a counter example.

Note: Each of a, b and c is a nonzero real number.