Lesson 56: Solving Radical Equations
A radical equation is an equation in which the variable appears under a radical sign. Solving radical equations usually involves isolating the radical, raising both sides to the appropriate power, and checking for extraneous solutions.
Steps to Solve Radical Equations
- Isolate the radical expression on one side of the equation.
- Raise both sides of the equation to the power that eliminates the radical.
- Solve the resulting equation.
- Check all solutions in the original equation to discard any extraneous solutions.
Examples
Example 1: √(x + 3) = 5
- Square both sides: (√(x + 3))² = 5² → x + 3 = 25
- Solve: x = 22
- Check: √(22 + 3) = √25 = 5 ✔
Example 2: √(2x − 1) − 3 = 0
- Isolate radical: √(2x − 1) = 3
- Square both sides: 2x − 1 = 9 → 2x = 10 → x = 5
- Check: √(2*5 − 1) − 3 = √9 − 3 = 3 − 3 = 0 ✔
Example 3 (Extraneous Solution): √(x + 4) = x − 2
- Square both sides: x + 4 = (x − 2)² → x + 4 = x² − 4x + 4
- Solve quadratic: 0 = x² − 5x → x(x − 5) = 0 → x = 0 or x = 5
- Check: x = 0 → √4 = 0 − 2 → 2 ≠−2 ⌠(extraneous)
- Check: x = 5 → √9 = 5 − 2 → 3 = 3 ✔
- Solution: x = 5
Practice Problems
- √(x + 7) = 4
- √(3x − 2) = 5
- √(2x + 5) + 3 = 8
- √(x − 1) = x − 3
- √(4x + 1) − 2 = 0