Find roots of absolute equation

Question

I am not sure how to solve the below equation for $x$

$2|x(x-x_0)|-|x(x-2x_0)|-|(x-2x_0)(x-x_0)|=0$

Where $x_0$ is a real number

Answer 1

The only way to solve this type of questions is to take cases on position of $x$ and start removing the absolute value.

Example:

If: $x>x_0$ && $x<2x_0$

$=> 2x(x-x_0)-x(2x_0-x)-(x-x_0)(2x_0-x)=0$

Solve this and see if the solution lies between the specified conditions.. or else reject the solution and check next case.

Answer 2

Hint: When solving for absolute value involved equations, most likely you need to separate the original equation into several equations when different domains, so that absolute value bars can be eliminated and the usual equation solving process follows.

The domains would be $x \gt 0$, $0 \ge x \gt x_0$, $x - x_0 \ge x \gt x - 2 x_0$, $x \ge x - 2 x_0$.

Remember the check if the answer is contradictory to the domain constrains. If it is, it means there exist no solution to that domain.