Solve $x^2 = 81$. List all real solutions.

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05-11-2026
Mathematics

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Solve $x^2 = 81$. List all real solutions.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T16:46:21-05:00

Identify the given equation as $x^2 = 81$.
Apply the square root property which states that if $x^2 = c$, then $x = \pm\sqrt{c}$.
Calculate the square root of 81, which is 9.
Conclude that the real solutions are $x = 9$ and $x = -9$, written as $\boxed{9, -9}$.

Explanation

Problem Analysis

The given equation is $x^2 = 81$, where $x$ is a real number. We aim to find all values of $x$ that satisfy this equation. The expression involves squaring an unknown value to result in a positive constant, 81.

Solving for x

To solve an equation of the form $x^2 = c$, where $c \ge 0$, we use the property that the solutions are given by $x = \sqrt{c}$ and $x = -\sqrt{c}$. In our specific case, $c = 81$. Therefore, we need to find the square root of 81. Since $9 \times 9 = 81$, the positive square root is $9$. Consequently, the solutions are $x = 9$ and $x = -9$.

Examples

Solving basic power equations is essential in geometry and physics. For example, if you know the area of a square is 81 square meters, the equation for the side length $s$ is $s^2 = 81$. Understanding that a negative root also exists helps in more advanced contexts where orientation or direction might be represented by negative numbers, though for physical lengths, we typically only consider the positive root.