Understanding Radical Equations

A Radical Equation is an equation in which the variable appears under a radical sign (usually a square root, but it could be any root, such as a cube root, fourth root and so on).

Steps to Solve Radical Equations:

  1. Isolate the Radical: Get the radical expression by itself on one side of the equation.
  2. Eliminate the Radical: Raise both sides of the equation to the power that will eliminate the radical.
  3. Solve the Resulting Equation: After eliminating the radical, solve the resulting equation.
  4. Check for Extraneous Solutions: Substitute your solutions back into the original equation to make sure they are valid.

Examples

Example 1: Square Root Radical Equation

Equation: $$ \sqrt{x + 3} = 5 $$

1. Isolate the radical: $$ \sqrt{x + 3} = 5 $$ (already isolated).

2. Eliminate the radical: Square both sides: $$ (\sqrt{x + 3})^2 = 5^2 $$.

$$ x + 3 = 25 $$

3. Solve for $ x $:

$$ x = 25 - 3 = 22 $$

4. Check the solution by substituting $$ x = 22 $$ back into the original equation:

$$ \sqrt{22 + 3} = \sqrt{25} = 5 $$

Since both sides are equal, $$ x = 22 $$ is a valid solution.

Example 2: Cube Root Radical Equation

Equation: $$ \sqrt[3]{2x - 1} = 3 $$

1. Isolate the radical: $$ \sqrt[3]{2x - 1} = 3 $$ (already isolated).

2. Eliminate the radical: Cube both sides: $$ (\sqrt[3]{2x - 1})^3 = 3^3 $$.

$$ 2x - 1 = 27 $$

3. Solve for $ x $:

$$ 2x = 27 + 1 = 28 $$

$$ x = \frac{28}{2} = 14 $$

4. Check the solution by substituting $$ x = 14 $$ back into the original equation:

$$ \sqrt[3]{2(14) - 1} = \sqrt[3]{28 - 1} = \sqrt[3]{27} = 3 $$

Since both sides are equal, $$ x = 14 $$ is a valid solution.

Remember, always check for extraneous solutions when solving radical equations!

Important notes: