Solving Radical Equations — Definition, Formula & Examples
Solving radical equations means finding the value of the variable trapped inside a radical (like a square root or cube root) by isolating the radical and then raising both sides of the equation to the appropriate power to eliminate it.
To solve a radical equation, one isolates the radical expression on one side of the equation, raises both sides to the index of the radical to produce a polynomial equation, solves that equation, and then verifies each candidate solution in the original equation to discard any extraneous solutions introduced by the powering step.
How It Works
Start by isolating the radical on one side of the equation. Then raise both sides to the power that matches the radical's index — square both sides for a square root, cube both sides for a cube root, and so on. This removes the radical and leaves you with a simpler equation to solve. After solving, substitute every answer back into the original equation. Any solution that makes the original equation false is extraneous and must be rejected.
Worked Example
Problem: Solve √(x + 3) = x − 3.
Square both sides: The radical is already isolated, so square both sides to eliminate the square root.
Rearrange into standard form: Move all terms to one side to get a quadratic equation.
Factor and solve: Factor the quadratic.
Check for extraneous solutions: Substitute x = 1: √(1+3) = 2 but 1−3 = −2. Since 2 ≠ −2, x = 1 is extraneous. Substitute x = 6: √(6+3) = 3 and 6−3 = 3. This checks out.
Answer: x = 6
Why It Matters
Radical equations appear throughout Algebra 2 and precalculus, especially in distance formulas, physics equations involving period of a pendulum, and geometry problems where you set a radical expression equal to a known length. Mastering the check step also builds the habit of verifying solutions, which is critical in higher math.
Common Mistakes
Mistake: Forgetting to check solutions in the original equation.
Correction: Squaring both sides can introduce extraneous solutions that satisfy the squared equation but not the original. Always substitute every answer back into the original radical equation before finalizing.