Radical Equations Examples

1. Radical Equation with a Radical on One Side and an Algebraic Expression on the Other Side

Example:

$$ \sqrt{2x + 7} = 3x - 5 $$

Steps to Solve:

  1. Isolate the Radical: The radical is already isolated on one side.
  2. Eliminate the Radical: Square both sides to remove the square root:
    $$ (\sqrt{2x + 7})^2 = (3x - 5)^2 $$
    $$ 2x + 7 = 9x^2 - 30x + 25 $$
  3. Solve the Resulting Quadratic Equation: Bring all terms to one side to form a quadratic equation:
    $$ 0 = 9x^2 - 32x + 18 $$
  4. Check for Extraneous Solutions: Solve the quadratic equation and check the roots in the original equation to ensure they are valid solutions.

2. Radical Equation with Two Term Radicals on One Side and an Algebraic Expression on the Other Side

Example:

$$ \sqrt{x + 3} + \sqrt{x - 2} = 4 $$

Steps to Solve:

  1. Isolate One of the Radicals: Move one radical to the other side:
    $$ \sqrt{x + 3} = 4 - \sqrt{x - 2} $$
  2. Eliminate the Radical: Square both sides to remove the first radical:
    $$ (\sqrt{x + 3})^2 = (4 - \sqrt{x - 2})^2 $$
    $$ x + 3 = 16 - 8\sqrt{x - 2} + (x - 2) $$
  3. Isolate the Remaining Radical: Combine like terms and isolate the radical:
    $$ 8\sqrt{x - 2} = 21 - 2x $$
  4. Eliminate the Radical Again: Square both sides again to eliminate the second radical:
    $$ 64(x - 2) = (21 - 2x)^2 $$
  5. Solve the Resulting Equation: Simplify and solve the resulting equation. Finally, check the solutions to ensure they are valid.

3. Radical Equation with a Quadratic Inside the Radical on One Side and an Algebraic Expression on the Other Side

Example:

$$ \sqrt{x^2 + 4x + 4} = x + 6 $$

Steps to Solve:

  1. Isolate the Radical: The radical is already isolated on one side.
  2. Eliminate the Radical: Square both sides to remove the square root:
    $$ (\sqrt{x^2 + 4x + 4})^2 = (x + 6)^2 $$
    $$ x^2 + 4x + 4 = x^2 + 12x + 36 $$
  3. Solve the Resulting Equation: Simplify and solve the linear equation:
    $$ 4x + 4 = 12x + 36 $$
    $$ -8x = 32 $$
    $$ x = -4 $$
  4. Check for Extraneous Solutions: Substitute \(x = -4\) back into the original equation to verify it is a valid solution.

4. Radical Equation Involving Higher Powers and More Complex Expressions

Example:

$$ \sqrt[3]{2x^2 - 3x + 7} = x + 2 $$

Steps to Solve:

  1. Isolate the Radical: The radical is already isolated on one side.
  2. Eliminate the Radical: Cube both sides to remove the cube root:
    $$ (\sqrt[3]{2x^2 - 3x + 7})^3 = (x + 2)^3 $$
    $$ 2x^2 - 3x + 7 = x^3 + 6x^2 + 12x + 8 $$
  3. Solve the Resulting Equation: Rearrange the terms to form a cubic equation:
    $$ x^3 + 4x^2 + 15x + 1 = 0 $$
  4. Solve the Cubic Equation: Use methods such as factoring, synthetic division, or numerical approaches to solve for \(x\).
  5. Check for Extraneous Solutions: Ensure that all solutions satisfy the original equation.

5. Radical Equation with Multiple Radicals on Different Sides

Example:

$$ \sqrt{3x + 1} = \sqrt{x + 7} $$

Steps to Solve:

  1. Eliminate the Radicals: Square both sides to remove both radicals:
    $$ (\sqrt{3x + 1})^2 = (\sqrt{x + 7})^2 $$
    $$ 3x + 1 = x + 7 $$
  2. Solve the Resulting Equation: Solve for \(x\):
    $$ 2x = 6 $$
    $$ x = 3 $$
  3. Check for Extraneous Solutions: Substitute \(x = 3\) back into the original equation to verify it is a valid solution.

6. Radical Equation with a Rational Expression Inside the Radical

Example:

$$ \sqrt{\frac{2x + 3}{x - 1}} = 4 $$

Steps to Solve:

  1. Isolate the Radical: The radical is already isolated on one side.
  2. Eliminate the Radical: Square both sides to remove the square root:
    $$ \frac{2x + 3}{x - 1} = 16 $$
  3. Solve the Resulting Rational Equation: Multiply both sides by \(x - 1\):
    $$ 2x + 3 = 16(x - 1) $$
    $$ 2x + 3 = 16x - 16 $$
  4. Simplify and Solve for \(x\):
    $$ 14x = 19 $$
    $$ x = \frac{19}{14} $$
  5. Check for Extraneous Solutions: Ensure the solution is valid by substituting it back into the original equation.