Surd

Surd

A surd is an irrational root that cannot be simplified into a whole number or a rational number.

For example:

$\sqrt{2}$ is a surd because it cannot be written exactly as a fraction.

$\sqrt{3}$, $\sqrt{5}$, and $\sqrt[3]{7}$ are also surds.

But:

$\sqrt{4} = 2$ is not a surd because it simplifies to a whole number.

Example of simplifying a surd:

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

Here, $\sqrt{3}$ is the surd part.

A surd is usually written in radical form using the root symbol.

Equation Containing Surds

An equation containing surds is an equation in which the unknown variable appears inside a surd (radical) expression.

Examples:

• $\sqrt{x} = 5$
• $\sqrt{x+4} = 7$
• $\sqrt{2x-1} + 3 = 8$
• $\sqrt{x} + \sqrt{x-9} = 9$

To solve such equations, the surd must be eliminated by using appropriate algebraic operations, usually by isolating the surd and squaring both sides of the equation.

Steps to Solve Equations Containing Surds

Step 1: Isolate the Surd

Move all non-surd terms to the other side of the equation so that the surd stands alone.

Step 2: Square Both Sides

Square both sides of the equation to remove the square root.

Step 3: Simplify

Expand and simplify the resulting equation.

Step 4: Solve the Equation

Find the value(s) of the variable.

Step 5: Check the Solution

Substitute the obtained value(s) into the original equation. Squaring both sides can introduce extraneous solutions, so verification is essential.