We are able to solve $x^2+4=0$ by square rooting both sides, but if we have $x^2=-4$ we can't solve. Firstly, why? Aren't they equal expressions?
Secondly, if we have $x^2=-4$, why can't we bring the four to the other side, square root it and then bring it back?
On
Whenever you solve by squaring you add extraneous invalid solutions.
For example if you "solve" $x = 5$ there is one solution. $x = 5$ that's it. If you square both sides you get $x^2 = 25$ so $x=\pm \sqrt 25 = \pm 5$ so the $x=5$ OR $x = -5$. But we know $x = 5$ and $x \ne -5$ so we added an extra value that isn't legitimate or acceptable.
This is because $x \to x^2$ is not one to one for every $k > 0$ there are two $x$ where $x^2 = k$ and we don't know which one is intended.
So when you say we can solve $x^2 + 4 = 0$ by squaring both sides.... well, no, we can not solve $x^2 + 4=0$ by squaring both sides.
If we square both sides we get $x^4 + 8x^2 + 16=0$ and .... well,, I have no idea how you thought you would solve this. But if you could solve it the solutions would be extraneous and you wouldn't know which one is correct.
Except... we do know which one.... we know $x^2 +4 = 0$ is the correct one and we know that $x^2 -4 = 0$ is the WRONG one.
That fact that $x^2+4=0$ is impossible and can not give us answer, and that $x^2 -4=0$ could give us an answer doesn't matter. It could give us an answer but we know it would be a wrong answer because we know $x^2 -4 = 0$ is not true.
......
Here's an anology: Suppose you knew that Profits or $-17$. You lost money. And your boss insists that you must find a way to show that profits were $35$. He doesn't care about truth he just cares that you give him the answer he wants.
But no matter what you do you have $p = -17$. And so you add $17$ to both sides:
$p + 17 = 0$. That must be true.
Then you multiply both sides by $p-35$.
$(p+17)(p-35) = 0\times (p-35)= 0$. That must be true.
SO $p^2 -18p - 595 = 0$. That must be true.
So using the quadratic equation you get
$p = \frac {18 + \sqrt {18^2 +4*595}}2$ or $p = \frac {18 - \sqrt{18^2 + 4*595}}2$. This must be true.
And $18^2 + 4*595 = 2704 = 52^2$ so $\sqrt{18^2 + 4*595} = 52$ so
$p = \frac {18+52}2 = \frac {70}2 = 35$ or $p = \frac {18-52}2 = \frac {-34}2 = -17$.
SO either $p = 35$ or $p = -17$. This must be true. You just don't know which one is true.
Anyway.... profits must be positive, you say, because otherwise they would be "loss", not profit.
SO $p = 35$. Your boss is happy.
(But you both end up in the street when reality bites the company in the ass.)
Actually there are a few assumptions we need to explicit. First of all we need to explicit where we are searching the solution for our equation. We can search for natural, integer, rational, real, complex solutions. In this case your question involves the real case.
Let's analize the real case. We can easily see that the problem has no solution because $x^2\geq 0$ and so $x^2+4\geq 4$. We can take the square root of both sides of the equation $x^2+4=0$. Before taking the square root we must assume that each side is nonnegative. This is true. Now we have the equation $\sqrt{x^2+4}=0$. Again this equation has no solution. If you consider the equation $x^2=-4$, you cannot take the square root of both sides, because one of them is nonnegative. Doing this you are going outside the real numbers and this bring us to the next case. When you are taking a square root in the real numbers you are always assuming that what will be under the root is nonnegative