How to find a substitution that transforms an equation into a particular equation. (Roots of polynomials). Find the value of $c$ so that the substitution $x=u+c$ transforms the equation $x^3-12x^2+45x-54=0$ into the equation $u^3-3u^2=0$
My first idea is to find the roots of $u^3-3u^2=0$ And getting $u=0$ and $u=3$, however I am not sure how to continue.
If someone could explain how to tackle this sort of problem, I would greatly appreciate it. Thanks.

Here's a relatively fast approach that avoids factoring a cubic.
Notice that $u^3 - 3 u^2 = u^2 (u - 3)$ has a double root at $u = 0$, so if $x = u + c$, the given polynomial, $$p(x) := x^3 − 12 x^2 + 45 x − 54$$ has a double root at $x = (0) + c = c$.
Since $c$ is a double root, it is also a root of $$p'(x) = 3 (x^2 - 8 x + 15) = 3 (x - 3) (x - 5) ,$$ so $c = 3$ or $c = 5$. But $5 \not\mid 54$, so the Rational Root Theorem implies that $p(5) \neq 0$, leaving
$$\color{#df0000}{\boxed{c = 3}} .$$
