Step by Step
1
Isolate the radical
Get the radical expression completely alone on one side of the equation before doing anything else.
2
Raise both sides to the index power
If it's a square root, square both sides; if it's a cube root, cube both sides — matching the radical's index eliminates it.
3
Solve the resulting equation
Once the radical is eliminated, solve the remaining equation using standard algebra steps.
4
Check every solution in the ORIGINAL equation
Squaring (or raising to any even power) can introduce extraneous solutions — answers that satisfy the squared equation but not the original radical equation. Every solution must be verified in the original equation before being accepted.
Applied Walkthrough
1
Solve √(x + 3) = x − 3. The radical is already isolated on the left.
2
Square both sides: x + 3 = (x − 3)², which expands to x + 3 = x² − 6x + 9.
3
Rearrange and solve: 0 = x² − 7x + 6, which factors to (x − 6)(x − 1) = 0, giving x = 6 or x = 1.
4
Check both in the ORIGINAL equation: x = 6 gives √9 = 3, which checks out. x = 1 gives √4 = 2, but the original right side would be 1 − 3 = −2 — that doesn't match, so x = 1 is extraneous and must be rejected.
Exam Application
Exams test specifically whether you check every solution in the original (unsquared) radical equation, since squaring both sides can introduce extraneous solutions that satisfy the squared version but not the original.
⚠ Common Trap
The single biggest trap in this topic is skipping the final check — assuming that any solution to the squared equation must also work in the original radical equation. It doesn't always; extraneous solutions are common and must be caught by checking.
✓ Quick Self-Check
1. What is the first step in solving a radical equation?
Isolate the radical completely on one side of the equation.
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2. Why must you check solutions in the original equation, not the squared one?
Because squaring both sides can introduce extraneous solutions that satisfy the squared equation but not the original.
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3. Solve √(x+3) = x−3 and identify any extraneous solution.
x = 6 is valid; x = 1 is extraneous and must be rejected.
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4. What operation do you perform to eliminate a square root from an equation?
Square both sides of the equation.
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5. What's the single biggest trap in solving radical equations?
Skipping the final check of each solution against the original (unsquared) equation.
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