Review of Algebra

RTE

Square Roots of Real Numbers

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

• Definition: The square root of a number a is a number x such that .

• Principal Square Root: The positive square root of a number is called the principal root and is denoted by the radical symbol .

• Negative Square Root: The negative square root is .

Example:

• To determine the square roots of 9, we look for numbers that satisfy .

• Testing :

• Testing :

• Therefore, the two square roots of 9 are 3 (principal root) and -3.

Key Properties:

• Every positive real number has two real square roots.

• The principal root is always non-negative.

• Zero has one square root: 0.

• Negative real numbers do not have real square roots (their roots are imaginary).

General Formula:

Applications:

• Solving quadratic equations

• Geometry (finding side lengths)

• Physics (calculating distances, speeds, etc.)