If the roots of the quadratic equation $x^2 − 8x + p^2$ = 0 are equal, then the positive value of $p$ is :

Given,

The roots of the quadratic equation $x^2 - 8x + p^2$ = 0 are equal.

Comparing the given equation with $ax^2 + bx + c$ = 0, we get :

$a = 1, b = -8 \text{ and } c = p^2$

We know that,

For equal roots, discriminant (D) = 0.

$$\begin{aligned}\Rightarrow b^2 - 4ac = 0 \\[1em] \Rightarrow (-8)^2 - 4 \times 1 \times p^2 = 0 \\[1em] \Rightarrow 64 - 4p^2 = 0 \\[1em] \Rightarrow 4p^2 = 64 \\[1em] \Rightarrow p^2 = 16 \\[1em] \Rightarrow p = \pm 4\end{aligned}$$

Since, the positive value of $p$ is required.

∴ $p = 4$.

Hence, option 3 is the correct option.