Roots equal but opposite in sign

For a quadratic equation $ax^2 + bx + c = 0$, let the roots be $\alpha$ and $-\alpha$.

Sum of roots $= \alpha + (-\alpha) = 0$.

Also, sum of roots $= -\frac{b}{a}$.

Thus, $-\frac{b}{a} = 0 \implies b = 0$.