Why can't I take the square root of both sides when finding the roots of a quadratic equation? Is $x^2 - x - 12 = 0$ equivalent to $x = \sqrt{x + 12}$?
I started with
$x^2 - x - 12 = 0$
and made the following changes:
$x^2 - x - 12 = 0$
$x^2 = x + 12$
$x = \sqrt{x + 12}$
From here I can eyeball it and see that x = 4 and x = -3 are solutions.
I know there is a better way to find the roots, but I was told that $x^2 - x - 12 = 0$ and $x = \sqrt{x + 12}$ are not equivalent. If they are not, why not?

It's not equivalent for two reasons:


*

*We don't know that $x + 12$ is non-negative, so it might not be valid to take square roots

*$\sqrt{x + 12}$ is always non-negative by definition, provided it's defined. So $x = -3$ is not actually a solution to this new equation.
