Using the derivative, find $a$ such that: $|x-a| = (x-2)^2$

So I've been studying Jay Cummings's Real Analysis book, and I've encountered this problem:Use the derivative to find all values of $ a $ such that the following holds: $$|x-a| = (x-2)^2 $$He doesn't explicitly mention that it should hold for all values of $x$. Although the solution is trivial: $$ a = x \pm (x-2)^2 $$ I really do not understand how come one uses the derivative for this.The reader is assumed to only know notions such as continuity and elementary set theory.