Bruce J. answered • 04/18/19
Caltech/Johns Hopkins Grad with 15+ years of tutoring experience
We notice from the fact that f(x) is quadratic and has a maximal value greater than zero that it has two real roots, which we will call t1 and t2. Consider the equation f(x) = 0, where f(x) = ax^2 +bx + c, a, b and c all real and a != 0 (which we can assume). We can factor the quadratic as follows:
ax^2 + bx + c = a(x^2 + (b/a)x + c/a) = a(x-t1)(x-t2), by definition of roots of a quadratic equation.
Expanding the third form of our quadratic above, we see that f(x) = a(x^2 - (t1 + t2)x + t1 * t2)), so that the coefficient b of x in our original equation is equal to b = -(t1 + t2)*a. The expression t1 + t2 is the sum of the roots that we're looking for. We also know that the maximum or minimum of a parabola f(x) = ax^2 + bx + c occurs at the value x = -b/2a, and we know that in our case that this is equal to -3. So....
(-b/2a) = -3 --> b = 6a ....
BUT
b = -(t1 + t2)*a from before....
SO
6a = -(t1 + t2)*a ---> (t1 + t2) = -6.
Hence the sum of the roots is -6. Notice that the maximal value of 5 did not matter.
