1962 AHSME Problems/Problem 34
For what real values of does have real roots?
Factoring, we get At this point, we could use the quadratic formula, but we're only interested in whether the roots are real, which occurs if and only if the discriminant () is non-negative. This function of is symmetric with respect to the y-axis. Plugging in for gives us , and it is clear that the function is strictly increasing over , and so it is positive for all real values of .