What is the difference between square root of x and plus or minus square root of x?

The square-root symbol means the principal square root, which is non-negative for x greater than or equal to zero. Plus or minus square root of x gives both solutions to the equation y squared equals x.

Square Roots • Radicals • Perfect Squares

Square Root Calculator

Q: What is a square root?

A square root of a number x is a number that gives x when multiplied by itself. If y squared equals x, then y is a square root of x.

Q: How do you simplify a square root?

Factor the number under the radical and remove the largest perfect-square factor. For example, square root of 72 equals square root of 36 times 2, which simplifies to 6 square root of 2.

Q: Can the square root of a negative number be calculated?

A negative number has no real square root, but it has an imaginary square root. For example, square root of negative 81 equals 9i, where i squared equals negative one.

Q: Is zero a perfect square?

Yes. Zero is a perfect square because zero squared equals zero, so the square root of zero is zero.

Q: Why does a positive number have two square roots?

A positive number has two square roots because both a positive number and its negative version square to the same positive value.

Use this Square Root Calculator to find the square root of any real number, simplify radicals when possible, identify perfect squares, and understand what the answer means. Enter a number, choose your rounding option, and the calculator will show the principal square root, the two real square-root solutions when they exist, and the complex square-root result for negative numbers.

A square root answers one direct question: what number, when multiplied by itself, gives the original number? For example, \( \sqrt{49} = 7 \) because \( 7^2 = 49 \). This page gives you the calculator, the square root formula, step-by-step method, simplified radical rules, worked examples, negative square roots, perfect squares, common mistakes, and FAQs in one WordPress-ready section.

Calculate \( \sqrt{x} \) Simplify radicals Check perfect squares Handle negative inputs

Square root formula

Definition of square root

\[ y = \sqrt{x} \quad \Longleftrightarrow \quad y^2 = x \]

The principal square root \( \sqrt{x} \) is the non-negative square root when \( x \ge 0 \).

For an equation such as \( y^2 = x \), the real solutions are \( y = \pm\sqrt{x} \) when \( x > 0 \).

Calculate a square root

Enter any real number. If the number is positive, the calculator shows the principal square root and both real square-root solutions. If the number is negative, the calculator shows the imaginary square root using \( i \), where \( i^2 = -1 \).

Number \( x \)

Decimal places

Tip: Try values such as \( 64 \), \( 72 \), \( 0.25 \), and \( -81 \) to see perfect squares, simplified radicals, decimals, and imaginary square roots.

Result

Enter a value and press calculate.

What is a square root?

A square root is a number that produces a given value when it is multiplied by itself. If \( y^2 = x \), then \( y \) is a square root of \( x \). The symbol used for the principal square root is \( \sqrt{x} \). The word principal is important because a positive number usually has two real square roots: one positive and one negative. For example, \( 8^2 = 64 \) and \( (-8)^2 = 64 \), so both \( 8 \) and \( -8 \) are square roots of \( 64 \). However, the symbol \( \sqrt{64} \) means the principal square root, so \( \sqrt{64} = 8 \), not \( \pm 8 \).

Square roots appear throughout mathematics because squaring is one of the most common operations. They are used in algebra, geometry, coordinate geometry, trigonometry, physics, statistics, engineering, computer graphics, finance, and measurement problems. Any time a quantity is squared and you need to reverse that operation, square roots become relevant. For example, if the area of a square is known, the side length is found with a square root. If the distance between two coordinate points is needed, the distance formula uses a square root. If the standard deviation of a data set is calculated from variance, square roots are used again.

The square root calculator on this page is designed to be both fast and educational. It gives the decimal value, identifies whether the number is a perfect square, simplifies radical form when possible for whole numbers, and explains what happens when the number is negative. This matters because a calculator that only gives a decimal can hide the structure of the answer. For example, \( \sqrt{72} \approx 8.485 \), but the simplified radical form \( 6\sqrt{2} \) is often the cleaner exact answer in algebra.

Square root formulas and rules

The basic square root definition is built from the relationship between squaring and square rooting. Squaring takes a number and multiplies it by itself. Square rooting reverses that operation. The formula can be written as:

Square root definition

\[ y = \sqrt{x} \quad \Longleftrightarrow \quad y^2 = x \]

This means \( y \) is the principal square root of \( x \) if \( y \) is non-negative and \( y^2 = x \). For positive values of \( x \), the equation \( y^2 = x \) has two real solutions:

Solutions to a square equation

\[ y^2 = x \quad \Longrightarrow \quad y = \pm\sqrt{x} \]

This distinction is one of the most common sources of mistakes. The expression \( \sqrt{x} \) means the principal square root. The equation \( y^2 = x \) asks for all values of \( y \) that square to \( x \), so it requires \( \pm\sqrt{x} \) when \( x > 0 \). For example, \( \sqrt{25} = 5 \), but the equation \( y^2 = 25 \) has solutions \( y = 5 \) and \( y = -5 \).

Rule	Formula	Important condition
Principal square root	\( \sqrt{x} \ge 0 \)	For real numbers, \( x \ge 0 \)
Square-root equation	\( y^2 = x \Rightarrow y = \pm\sqrt{x} \)	For two real solutions, \( x > 0 \)
Product rule	\( \sqrt{ab} = \sqrt{a}\sqrt{b} \)	Works cleanly for \( a \ge 0 \), \( b \ge 0 \)
Quotient rule	\( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)	For \( a \ge 0 \), \( b > 0 \)
Perfect square	\( \sqrt{m^2} = \|m\| \)	The result is non-negative
Negative square root	\( \sqrt{-x} = i\sqrt{x} \)	For \( x > 0 \), using \( i^2 = -1 \)

How to calculate a square root step by step

There are different ways to calculate a square root depending on the type of answer you need. For quick numerical work, a calculator gives the decimal approximation immediately. For algebra, simplified radical form is often more useful. For equations, you must decide whether the question asks for the principal square root or all square-root solutions. The following steps cover the standard method used by this page.

Identify the number under the radical. In \( \sqrt{x} \), the number \( x \) is called the radicand. For example, in \( \sqrt{72} \), the radicand is \( 72 \).
Check whether the radicand is positive, zero, or negative. If \( x > 0 \), the principal square root is positive and the square-root equation has two real solutions. If \( x = 0 \), the square root is \( 0 \). If \( x < 0 \), the real square root does not exist, but the complex square root can be written using \( i \).
Check whether the number is a perfect square. A perfect square has a whole-number square root. For example, \( 81 \) is a perfect square because \( 9^2 = 81 \).
Simplify the radical if possible. If the number is not a perfect square, factor out the largest perfect-square factor. For example, \( 72 = 36 \times 2 \), so \( \sqrt{72} = \sqrt{36}\sqrt{2} = 6\sqrt{2} \).
Calculate the decimal approximation. If an exact radical is not convenient, use a decimal approximation. For example, \( \sqrt{72} \approx 8.485 \).
Include both signs when solving an equation. If the problem is \( y^2 = 72 \), then \( y = \pm\sqrt{72} = \pm 6\sqrt{2} \). If the problem is simply \( \sqrt{72} \), the principal value is \( 6\sqrt{2} \).

Important distinction: \( \sqrt{36} = 6 \), but \( x^2 = 36 \) gives \( x = \pm 6 \). The square-root symbol gives the principal value; the equation gives both solutions.

Worked examples

Example 1: Find \( \sqrt{64} \)

The number \( 64 \) is a perfect square because \( 8 \times 8 = 64 \). Therefore:

\[ \sqrt{64} = 8 \]

If the question is \( x^2 = 64 \), then the solutions are:

\[ x = \pm 8 \]

The interpretation is that \( 8 \) is the principal square root, while \( 8 \) and \( -8 \) are both solutions to the equation \( x^2 = 64 \).

Example 2: Simplify \( \sqrt{72} \)

The number \( 72 \) is not a perfect square, but it contains a perfect-square factor. The largest perfect-square factor of \( 72 \) is \( 36 \). So:

\[ \sqrt{72} = \sqrt{36 \times 2} \] \[ \sqrt{72} = \sqrt{36}\sqrt{2} \] \[ \sqrt{72} = 6\sqrt{2} \]

The decimal value is:

\[ \sqrt{72} \approx 8.485 \]

Both \( 6\sqrt{2} \) and \( 8.485 \) represent the same principal square root, but \( 6\sqrt{2} \) is exact while \( 8.485 \) is rounded.

Example 3: Find \( \sqrt{0.25} \)

A decimal can also have a clean square root. Since \( 0.5^2 = 0.25 \), we have:

\[ \sqrt{0.25} = 0.5 \]

If the equation is \( x^2 = 0.25 \), then:

\[ x = \pm 0.5 \]

Example 4: Find \( \sqrt{-81} \)

A negative number does not have a real square root because no real number squared gives a negative result. However, using the imaginary unit \( i \), where \( i^2 = -1 \), we can simplify:

\[ \sqrt{-81} = \sqrt{81}\sqrt{-1} \] \[ \sqrt{-81} = 9i \]

If the equation is \( x^2 = -81 \), then the two complex solutions are:

\[ x = \pm 9i \]

Perfect squares and square roots

A perfect square is a number that can be written as a whole number multiplied by itself. For example, \( 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 \) are perfect squares because they equal \( 1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2, 10^2 \). Recognizing perfect squares helps you simplify radicals quickly. It also helps with algebra, geometry, factoring, quadratic equations, and mental math.

Perfect squares are especially useful when simplifying square roots. The goal is to split the radicand into a perfect-square factor and a leftover factor. For example, \( 50 = 25 \times 2 \), so \( \sqrt{50} = 5\sqrt{2} \). Similarly, \( 98 = 49 \times 2 \), so \( \sqrt{98} = 7\sqrt{2} \). This method keeps answers exact.

Number	Perfect-square factor	Simplified square root	Approximate decimal
\( 12 \)	\( 4 \times 3 \)	\( 2\sqrt{3} \)	\( 3.464 \)
\( 18 \)	\( 9 \times 2 \)	\( 3\sqrt{2} \)	\( 4.243 \)
\( 20 \)	\( 4 \times 5 \)	\( 2\sqrt{5} \)	\( 4.472 \)
\( 45 \)	\( 9 \times 5 \)	\( 3\sqrt{5} \)	\( 6.708 \)
\( 75 \)	\( 25 \times 3 \)	\( 5\sqrt{3} \)	\( 8.660 \)
\( 108 \)	\( 36 \times 3 \)	\( 6\sqrt{3} \)	\( 10.392 \)

Negative square roots and imaginary numbers

In the real number system, square roots of negative numbers are not defined because the square of any real number is non-negative. A positive number squared is positive, a negative number squared is also positive, and zero squared is zero. Therefore, no real number has a square equal to \( -9 \), \( -16 \), or \( -81 \).

To handle negative square roots, mathematics uses the imaginary unit:

\[ i = \sqrt{-1} \] \[ i^2 = -1 \]

This allows negative square roots to be simplified. The rule is:

\[ \sqrt{-x} = i\sqrt{x} \]

For example:

\[ \sqrt{-49} = 7i \] \[ \sqrt{-20} = 2i\sqrt{5} \]

The calculator above follows this rule. If you enter a negative number, it returns the principal imaginary square root and also shows the two complex solutions for the equation \( y^2 = x \). This is useful for algebra, quadratic equations, complex numbers, and advanced mathematics where imaginary values are expected.

Where square roots are used

Square roots are not just an isolated arithmetic topic. They appear in many practical and academic contexts because they reverse squared relationships. In geometry, the side length of a square is the square root of its area. If a square has area \( A \), then its side length is:

\[ s = \sqrt{A} \]

In coordinate geometry, the distance between two points uses a square root:

\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]

In right-triangle problems, the Pythagorean theorem often requires a square root when solving for a missing side:

\[ c = \sqrt{a^2+b^2} \]

In statistics, the standard deviation is the square root of variance:

\[ \sigma = \sqrt{\sigma^2} \]

In physics and engineering, square roots occur in formulas involving speed, energy, oscillation, signal strength, and measurement. In computer graphics, square roots help calculate distances, vector lengths, lighting effects, and movement. In finance and risk analysis, square roots appear in volatility and standard deviation calculations. Learning square roots carefully builds a foundation for many later topics.

Common mistakes with square roots

Writing \( \sqrt{25} = \pm 5 \). This is not correct. The principal square root is \( \sqrt{25} = 5 \). The equation \( x^2 = 25 \) has solutions \( x = \pm 5 \).
Forgetting to simplify radicals. A decimal answer may be acceptable in some contexts, but algebra often expects exact simplified form. For example, \( \sqrt{72} \) should be simplified to \( 6\sqrt{2} \).
Splitting sums incorrectly. In general, \( \sqrt{a+b} \ne \sqrt{a}+\sqrt{b} \). For example, \( \sqrt{9+16} = \sqrt{25} = 5 \), but \( \sqrt{9}+\sqrt{16} = 3+4 = 7 \).
Ignoring negative radicands. A negative number has no real square root, but it can have imaginary square roots. For example, \( \sqrt{-36} = 6i \).
Rounding too early. If you round intermediate values before finishing a problem, your final answer may become less accurate. Keep exact radical form when possible and round only at the end.
Assuming every square root is irrational. Perfect squares have whole-number square roots, and some decimals and fractions have simple square roots too. For example, \( \sqrt{0.25} = 0.5 \) and \( \sqrt{\frac{1}{4}} = \frac{1}{2} \).

FAQ

What is a square root?

A square root of a number \( x \) is a number that gives \( x \) when multiplied by itself. If \( y^2 = x \), then \( y \) is a square root of \( x \).

What is the difference between \( \sqrt{x} \) and \( \pm\sqrt{x} \)?

The symbol \( \sqrt{x} \) means the principal square root, which is non-negative for \( x \ge 0 \). The expression \( \pm\sqrt{x} \) gives both solutions to the equation \( y^2 = x \).

How do you simplify a square root?

Factor the number under the radical and remove the largest perfect-square factor. For example, \( \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2} \).

Can the square root of a negative number be calculated?

A negative number has no real square root, but it has an imaginary square root. For example, \( \sqrt{-81} = 9i \), where \( i^2 = -1 \).

Is zero a perfect square?

Yes. Zero is a perfect square because \( 0^2 = 0 \), so \( \sqrt{0} = 0 \).

Why does a positive number have two square roots?

A positive number has two square roots because both a positive number and its negative version square to the same positive value. For example, \( 7^2 = 49 \) and \( (-7)^2 = 49 \).

Related tools and guides

Square roots connect naturally to exponents, radicals, equations, geometry, and scientific calculation. Use these related Num8ers resources when you want to continue building the same skill set.