For what value of $a$ does the equation $|x(x-4)|=a$ have exactly 3 real solutions? I simplified the equation and got two cases where $$x(x-4)=a$$
and 
$$x(x-4)=-a.$$
How do I go after that. I know how you could get $2$ solutions or $4$ solutions, but how do you get $3$ solutions?

The expression inside the absolute value signs is a quadratic, so the graph of $y=x(x-4)$ is a parabola. Its vertex is at $(2,-4)$, so the equation $x(x-4)=-4$ has precisely $1$ solution. Meanwhile, the equation $x(x-4)=4$ has two solutions. Putting these together, you should be able to find the answer to your question.
