On The Consistency of Equations

Question

There's a certain mathematical "phenomenon" that I don't seem to grasp. Presented the equation: $x^2 + 4x + 4 = 0$, we can readily find that the equation has exactly 1 solution ($x = –2$). However, when we isolate $x^2$ and take the square root of both sides, we get the equality $x = \sqrt{–4x – 4}$. However, this equality has zero roots. I don't quite understand why, though. I previously thought that any root of the first equation must also be a root of the second. If not, then when is it the case that roots are transfered from one equation to another? Thank you in advance.

Answer 1

COMMENT: You can note that the equation $x^2+4x+4=0$ is equivalent to $(x+2)^2=0$ so $x=-2$ is the only solution (it is a double root so "two" roots because it is a quadratic equation of one unknown over a field, which is $\mathbb Q$. On the other hand $x^2+4x+4=0$ is equivalent to $x=\pm\sqrt{-4x-4}$ and it is also an equation to be solved. We can see that the only solution is $x=-2$ (It could not be otherwise!) and the signe "+" must be discarded because, simply, it does not satisfy the equation. (This is usual in a lot of problems).