Find the least integral value of $k$ for which the equation $x^2 - 2 \left(k + 2\right)x + 12 + k^2 = 0$ has two different real roots.
Given quadratic equation: $\;\;\;$ $x^2 - 2 \left(k + 2\right) x + 12 + k^2 = 0$ $\;\;\; \cdots \; (1)$
Equation $(1)$ has two different real roots when its discriminant $\Delta > 0$
i.e. $\;$ $\Delta = \left[-2 \left(k + 2\right)\right]^2 - 4 \times 1 \times \left(12 + k^2\right) > 0$
i.e. $\;$ $4 \left(k^2 + 4k + 4\right) - 48 - 4k^2 >0$
i.e. $\;$ $4k^2 + 16k + 16 - 48 - 4k^2 > 0$
i.e. $\;$ $16 k > 32$
i.e. $\;$ $k > 2$
$\therefore \;$ The least value of $k$ for which equation $(1)$ has two different real roots is $k = 3$