Section 5.3: Radical Equations
This section covers equations involving radicals and demonstrates techniques for isolating radicals and solving for the variable, including checking for extraneous solutions.
Example 1: Solving a Simple Radical Equation
Solve \( \sqrt{x+5} = 7 \).
Step 1: Square both sides: \( x + 5 = 49 \)
Step 2: Solve for x: \( x = 44 \)
Example 2: Radical Equation with Variable on Both Sides
Solve \( \sqrt{2x+3} = x-1 \).
Step 1: Square both sides: \( 2x + 3 = (x-1)^2 = x^2 - 2x + 1 \)
Step 2: Rearrange: \( 0 = x^2 - 4x - 2 \)
Step 3: Solve quadratic: \( x = 2 \pm \sqrt{6} \)
Step 4: Check for extraneous solutions → only valid solution: \( x = 2 + \sqrt{6} \)
Practice Problems
- Solve \( \sqrt{x+9} = 5 \)
- Solve \( \sqrt{3x-2} = x+1 \)
- Solve \( \sqrt{2x+7} - 3 = 0 \)
- Solve \( \sqrt{x+4} + \sqrt{x} = 6 \)
- Check for extraneous solutions in \( \sqrt{5x+1} = x+3 \)