Three Distinct Real Roots of Absolute Quadratic

Let $g(x) = x^2 - 4x - 13 = (x-2)^2 - 17$. The vertex of the parabola $g(x)$ is at $(2, -17)$.

The graph of $f(x) = |x^2 - 4x - 13|$ flips the negative portion of $g(x)$ above the x-axis. The local maximum of $f(x)$ occurs at the vertex $x = 2$, where $f(2) = |-17| = 17$.

For $f(x) = r$ to have exactly three distinct roots, the horizontal line $y = r$ must touch the local maximum of the inverted parabola. Thus, $r = 17$.