Square and square root

To square a number means to multiply it by itself, like $5^2 = 25$.
A square is never negative, because a positive times a positive and a negative times a negative are both positive.
So $(x + 1)^2 \geq 0$ for every value of $x$.

A square root undoes a square.
When you solve an equation like $x^2 = 16$, you must allow both signs, because $4^2 = 16$ and $(-4)^2 = 16$ are both true.
So $x = \pm\sqrt{16} = \pm 4$.

Be careful: the square-root symbol $\sqrt{\phantom{x}}$ on a known positive number means only the positive root.
So $\sqrt{4} = 2$, not $\pm 2$.
The two signs appear only when you are solving for a variable that was squared.

A square root of any real expression is also $\geq 0$, so $\sqrt{2x + 3} \geq 0$ whenever it is defined.
And an equation like $x^2 = -5$ has no real solution, because a square can never be negative.

This idea helps when rearranging formulas.
For example, from $A = \pi r^2$ you divide by $\pi$ and take the square root to get $r = \sqrt{\frac{A}{\pi}}$.
When the variable stands for a real quantity like a length or speed, you keep only the positive root.

Solve $x^2 = 49$.
Take the square root of both sides, keeping both signs: $x = \pm\sqrt{49}$.
So $x = 7$ or $x = -7$.

What is $\sqrt{81}$?
The square-root symbol on a known positive number gives only the positive root.
So $\sqrt{81} = 9$, not $\pm 9$.

The area of a circle is $A = \pi r^2$. Solve for the radius $r$.
Divide both sides by $\pi$: $\frac{A}{\pi} = r^2$.
Take the square root, and since a radius is a length we keep the positive root: $r = \sqrt{\frac{A}{\pi}}$.