Can you take the square root of a negative number?

No, not in the real number system. Any real number (positive or negative) squared gives a positive result, so a negative number cannot be a perfect square. In advanced mathematics, you can take square roots of negative numbers using imaginary numbers, but at this level, we work only with non-negative radicands.

Why do we write \pm when solving equations but not when we see \sqrt{4}?

Because the square root symbol \sqrt{} by definition means just the non-negative root—so \sqrt{4} = 2 only. But when solving an equation like x^2 = 4, both x = 2 and x = -2 work, so we write x = \pm 2. The symbol and the equation have different meanings.

What does it mean if the answer is not a whole number, like \sqrt{7}?

It means 7 is not a perfect square, so its square root is irrational (it cannot be written as a fraction, and its decimal never ends or repeats). You can either leave it as \sqrt{7} (exact form) or calculate \sqrt{7} \approx 2.646 (decimal approximation).

How is squaring different from square root?

Squaring means multiplying a number by itself (e.g., 3^2 = 9), while square root is the reverse—finding the number that was squared (e.g., \sqrt{9} = 3). They undo each other: if you square a number and then take its square root, you get back what you started with (for non-negative numbers).

Square Root

Square root finds the non-negative number that, when multiplied by itself, gives your starting number; use it to undo squaring in equations.

\sqrt{a} = b \iff b^2 = a, ; b \ge 0

Solve a problem with Square Root

Type the problem. The solver will use Square Root where Square Root is the right tool, and tell you when it is not.

What each symbol means

What Square Root takes

Symbol	Meaning
$a$	The radicand — the number under the square root sign; it must be non-negative (zero or positive) because no real number squared gives a negative result.
$b$	The result of the square root — always non-negative by definition; when $b^2 = a$ , this is the principal (positive) square root.

When to use it

Reach for square root when you need to find what number, when multiplied by itself, produces a given value.

Level

Usually taught in: Pre-Algebra

Worked examples

1. Find a perfect square

Problem

Find

\sqrt{25}

$n \cdot n = 25$
We're looking for a number $n$ that when multiplied by itself equals 25.
$5 \cdot 5 = 25$
We find that 5 multiplied by 5 equals 25, so $n = 5$ .
$\sqrt{25} = 5$
Therefore, the square root of 25 is 5.

Answer: $\sqrt{25} = 5$

For perfect squares, we find the square root by recognizing which number multiplied by itself gives the original number. Here, we see that $5 \times 5 = 25$ , so $\sqrt{25} = 5$ .

2. Solve a quadratic equation

Problem

Solve

x^2 = 7

for

x

$x^2 = 7$
We need to find what value of $x$ , when squared, gives 7.
$x = \pm \sqrt{7}$
Taking the square root of both sides, we get $x = \pm\sqrt{7}$ ; the $\pm$ appears because both positive and negative values square to give 7.
$x \approx \pm 2.646$
Calculating the decimal form: $\sqrt{7} \approx 2.646$ , so $x \approx 2.646$ or $x \approx -2.646$ .

Answer: $x = \pm \sqrt{7} \approx \pm 2.646$

When solving an equation where $x$ is squared, we use square root to find $x$ . The $\pm$ symbol is crucial: it reminds us that both the positive and negative values satisfy the original equation, since both positive and negative numbers squared can give a positive result.

3. Find a side length from area

Problem

A coffee shop sells square napkins with an area of 36 square inches. What is the side length of one napkin?

$A = s^2$
For a square, the area is the side length $s$ multiplied by itself: $A = s^2$ .
$36 = s^2$
Since the area is 36 square inches, we have $s^2 = 36$ .
$s = \sqrt{36}$
To find the side length, we take the square root of both sides.
$s = 6$
We calculate $\sqrt{36} = 6$ because $6 \times 6 = 36$ .

Answer: $s = 6 \text{ inches}$

Square root is perfect for this problem because it reverses the operation of squaring. Since area is calculated as side length squared, we use square root to find the side length from the area. This real-world application shows why square root is essential in geometry.

Common mistakes

Where Square Root usually goes wrong

The mistake	Why it is wrong	The fix
Writing $\sqrt{4} = \pm 2$	By definition, the square root symbol always gives the non-negative (principal) square root, not both positive and negative values.	$\sqrt{4} = 2$ only; if you need both roots, write the equation as $x^2 = 4$ , so $x = \pm 2$ .
Taking the square root of only one term in an addition, like saying $\sqrt{9 + 16} = \sqrt{9} + \sqrt{16} = 3 + 4 = 7$	Square root does not distribute over addition or subtraction, so you cannot split the radicand apart and take roots separately.	$\sqrt{9 + 16} = \sqrt{25} = 5$ , not 7; always simplify what's under the radical before taking the square root.
Thinking that $\sqrt{-9}$ equals $-3$ or that negative radicands are okay	The square root of a negative number is not a real number because no real number squared produces a negative result.	$\sqrt{-9}$ is undefined in the real number system; square roots only work with non-negative radicands.

Tips and when to use something else

Use square root to undo squaring: if you see $x^2 = 16$ , take the square root of both sides to get $x = \pm 4$ .
Always check: multiply your answer by itself to verify it gives the number under the radical.
For cubic equations like $x^3 = 8$ , reach for Cube Root instead of Square Root—it's the operation that undoes cubing.
Remember $\sqrt{x^2} = |x|$ (use Absolute Value), not just $x$ , because square root always returns a non-negative result.

Related formulas

Absolute ValueAbsolute value measures the distance a number is from zero without regard to direction, so it is always positive or zero.Cube RootCube root finds the number that, when multiplied by itself three times, equals your original number—the inverse of cubing.Order of OperationsOrder of Operations is the set of rules that tells you which calculation to do first when you have multiple operations in one expression.Prime FactorizationPrime factorization breaks any whole number into a unique multiplication of primes—essential for finding common factors and simplifying fractions.

Frequently asked questions

Can you take the square root of a negative number?: No, not in the real number system. Any real number (positive or negative) squared gives a positive result, so a negative number cannot be a perfect square. In advanced mathematics, you can take square roots of negative numbers using imaginary numbers, but at this level, we work only with non-negative radicands.
Why do we write $\pm$ when solving equations but not when we see $\sqrt{4}$ ?: Because the square root symbol $\sqrt{}$ by definition means just the non-negative root—so $\sqrt{4} = 2$ only. But when solving an equation like $x^2 = 4$ , both $x = 2$ and $x = -2$ work, so we write $x = \pm 2$ . The symbol and the equation have different meanings.
What does it mean if the answer is not a whole number, like $\sqrt{7}$ ?: It means 7 is not a perfect square, so its square root is irrational (it cannot be written as a fraction, and its decimal never ends or repeats). You can either leave it as $\sqrt{7}$ (exact form) or calculate $\sqrt{7} \approx 2.646$ (decimal approximation).
How is squaring different from square root?: Squaring means multiplying a number by itself (e.g., $3^2 = 9$ ), while square root is the reverse—finding the number that was squared (e.g., $\sqrt{9} = 3$ ). They undo each other: if you square a number and then take its square root, you get back what you started with (for non-negative numbers).

Need a different method?

The full solver is not scoped to one formula — type any problem and it will pick the method.

Open the math solver

Reviewed 2026-09-18