Explanation

(c)
Given, $(x - a)(x - a - 1) + (x - a - 1)(x - a - 2) + (x - a)(x - a - 2) = 0$
Let $x - a = t$, then
$t(t - 1) + \ (t - 1)(t - 2) + t(t - 2) = 0$
$\Rightarrow t^{2} - t + t^{2} - 3t + 2 + t^{2} - 2t = 0$
$\Rightarrow \ \ \ 3t^{2} - 6t + 2 = 0$
$\Rightarrow t = \frac{6 \pm \sqrt{36 - 24}}{2(3)} = \frac{6 \pm 2\sqrt{3}}{2(3)}$
$\Rightarrow x - a = \frac{3 \pm \sqrt{3}}{3}$
$\Rightarrow \ \ x = a + \frac{3 \pm \sqrt{3}}{3}$
Hence, $x$ is real and distinct