TITLE: Square root in quadratic equation
QUESTION [2 upvotes]: I ran into a Math problem in which I was told to solve like that:
\begin{align}
x^ 2 - 25 &= 0 \\
x^2 & = 25 \\
x & = \sqrt{25}\\
x & = 5
\end{align}
I wonder what the logical and also intuictive explanation might be on removing the power of x and on the other hand, adding a square root to the other side.
Could you guys help me out on this? You see, even though I did solve the problem, I think it is only worthy if I understand the logical explanation behind it.

REPLY [0 votes]: I understand that where you are struct, if I am not wrong. Is the following your question? 
$$ x^2 = 25 \implies x =5 ?$$
What is square root of $y$?
When you write $\sqrt{y} = \sqrt[2]{y}$, you mean that you are searching a number $x$ such that $x^2 = y$. 
More generally, when we have $\sqrt[n]{y}$, it is nothing but a number $x$ such that $x^n = y$. 
Having the above observations and notations, 
$$x^2 = {25} \implies x = \sqrt{25} = \sqrt{5^2}  =\sqrt{(-5)^2} $$
which evidently proves that $x = 5$ or $x = -5$. 
If you are sure that $x$ is positive then $x = 5$.
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