For what values of b is -2 not in the range of the function f(x)=x^2+bx+2? Express your answer in interval notation.
We see that -2 is not in the range of f(x) = x^2 + bx + 2 if and only if the equation x^2 + bx + 2 = -2 has no real roots. We can re-write this equation as
x^2 + bx + 4 = 0. The discriminant of this quadratic is b^2 - 4 * 4 = b^2 - 16. The quadratic has no real roots if and only if the discriminant is negative, so , b^2 - 16 < 0 or b^2 < 16. The set of values that satisfy this inequality is (-4,4).
Also, this seems like an AoPS Alcumus Problem...
If it is... just letting you know that it's illegal to ask for alcumus problems, as AoPS has total rights over them.
Thanks!
