Find the values of the coefficient $a$ for which the curve $y = x^2 + ax + 25$ touches the $X$ axis.
When the given curve $\;\;$ $y = x^2 + ax + 25$ $\;$ touches the $X$ axis, its y coordinate $y = 0$
i.e. $\;$ $x^2 + ax + 25 = 0$ $\;\;\; \cdots \; (1)$
Since the given curve touches the $X$ axis (at one point)
$\implies$ equation $(1)$ should have two real equal roots
Now, for real equal roots of equation $(1)$,
its discriminant $= \Delta = a^2 - 4 \times 1 \times 25 = 0$
i.e. $\;$ $a^2 - 100 = 0$
i.e. $\;$ $a^2 = 100$
i.e. $\;$ $a = \pm 10$