Answer
Francesca was wrong because the quadratic formula works even when $b=0$.
Work Step by Step
RECALL:
The quadratic equation $ax^2+bx+c=0$ can be solved using the quadratic formula:
$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
The given equation has:
$a=1, b=0, c=-19$
Substitute these values into the quadratic formula to obtain:
$x=\dfrac{-0 \pm \sqrt{0^2-4(1)(-19)}}{2(1)}
\\x=\dfrac{\pm\sqrt{0+76}}{2}
\\x=\dfrac{\pm\sqrt{76}}{2}
\\x=\dfrac{\pm\sqrt{4(19)}}{2}
\\x=\dfrac{2\sqrt{19}}{2}
\\x=\pm\sqrt{19}$
Thus, the solution set is $\color{blue}{\left\{-\sqrt{19}, \sqrt{19}\right\}}$.
Francesca was wrong because the quadratic formula actually works regardless of whether $b=0$ or not.
