Find $\;$ $p$ $\;$ in the equation $\;$ $x^2 - 4x + p = 0$ $\;$ if it is known that the sum of the squares of its roots is equal to $16$.
Given quadratic equation: $\;\;\;$ $x^2 - 4x + p = 0$
Relation between the roots $\alpha$ and $\beta$ of the given quadratic equation is
$\alpha^2 + \beta^2 = 16$ $\;\;\; \cdots \; (1)$
Sum of roots of the given quadratic equation: $\;$ $\alpha + \beta = 4$ $\;\;\; \cdots \; (2a)$
Product of roots of the given quadratic equation: $\;$ $\alpha \cdot \beta = p$ $\;\;\; \cdots \; (2b)$
Squaring equation $(2a)$ gives
$\left(\alpha + \beta\right)^2 = 4^2$
i.e. $\;$ $\alpha^2 + \beta^2 + 2 \alpha \beta = 16$ $\;\;\; \cdots \; (3)$
In view of equations $(1)$ and $(2b)$, equation $(3)$ becomes
$16 + 2p = 16$
i.e. $\;$ $2p = 0$ $\implies$ $p = 0$