Identify the condition for equal roots of a quadratic equation.

For a quadratic equation in the form $Ax^2 + Bx + C = 0$, the roots are equal if and only if the discriminant is zero. The discriminant is given by the formula $\Delta = B^2 - 4AC$.

The given quadratic equation is $ax^2 + bx + c = 0$.
Comparing this to the standard form, we have $A = a$, $B = b$, and $C = c$.
Set the discriminant to zero:

Add $4ac$ to both sides of the equation:

Divide both sides by $4a$ to isolate $c$:

The calculated value of $c$ is $\frac{b^2}{4a}$, which matches option (B).

The final answer is $\text{(B)}$.