Question

How do i check an equation for extraneous solutions?

Answer 1

Example:

$\sqrt x(x+1)=0$

This suggests two solutions, $x=0$ and $x=-1$.

However, upon plugging these solutions back into the equation, you get

$\sqrt0(0+1)=0(1)=0$

which checks out, but

$\sqrt{-1}(1-1)=0$

does not because $\sqrt x$ is defined only for $x\ge0$ (assuming you're looking for real solutions only). So, we call $x=-1$ an extraneous solution, and the complete solution set (over the real numbers) is $x=0$.

Answer 2

An extraneous solution is a solution that arises from the solving process that is not really a solution at all! Specifically, whenever you raise both sides of an equation to an even power, you must check for extraneous solutions.