Square Roots and Their Properties

Square Roots

Definition and Properties

The square root of a number is a value that, when multiplied by itself, yields the original number. For any positive real number, there are always two square roots: one positive (the principal root) and one negative.

• Definition: If $x^2 = a$, then $x$ is a square root of $a$.

• Principal Root: The positive square root is called the principal root.

• Negative Root: The negative square root is also valid, since $(-x)^2 = x^2$.

Examples

• Example: To find the square roots of 9, solve $x^2 = 9$.

• Testing $x = 3$: $3^2 = 9$

• Testing $x = -3$: $(-3)^2 = 9$

• Therefore, the square roots of 9 are 3 and -3.

General Principle

• For any positive real number $a$, the equation $x^2 = a$ has two solutions: $x = \sqrt{a}$ and $x = -\sqrt{a}$.

• For example, $x^2 = 16$ yields $x = 4$ and $x = -4$.

Key Formula

• The square root of $a$ is denoted as $\sqrt{a}$.

• Both $\sqrt{a}$ and $-\sqrt{a}$ are square roots of $a$.

Additional info: This topic is foundational for College Algebra, especially in solving quadratic equations and understanding properties of real numbers.