Why is $(\sqrt{x^2})$ equal to $|x|$

Question

I don't understand why you have to write the absolute value sign when solving for the square root of $x$ squared.

Shouldn't the answer automatically be positive? Why is the absolute value sign needed?

Answer 1

$$\sqrt{x^2}=\begin{cases}x & \text{if} \: x\geq0 \\ -x & \text{if} \: x<0 \end{cases} = |x|$$

Is it clear?

Answer 2

The answer is automatically positive, but $x$ in the radicand isn't necessarily.

Another justification:

Both are non-negative, and their squares are equal.

Answer 3

Yes, the square root is automatically positive - that's exactly why we need the absolute value when we write $\sqrt{x^2}=|x|$. If instead we wrote $\sqrt{x^2}=x$ that would be wrong; for $x<0$ that would say the square root of $x^2$ was not positive.