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.

Playing with some graphs it seems this should always be true.  The graphs of y=[each graph] have the same y-intercept (0,c) from there they each curve up or down differently and since a/2 is between a and -a it ends up crossing the x-axis between them.

The three quadratic equations seems not to be the way to go, however since the algebra gets out of hand.  At one point I ended up with an unfactorable quartic in b.

Posted by Jer on 2013-02-17 19:19:06