Start with your attempted work. “Why did this extra root appear?” gives the explanation a specific target; asking for another full solution can hide the same mistake inside new wording.
Follow a radical equation through squaring
This original practice problem uses real numbers and the principal, nonnegative square root:
sqrt(x + 6) = x
The radical requires x + 6 ≥ 0. The original right side must also be nonnegative because it equals a principal square root, so any solution must have x ≥ 0.
Squaring both sides gives x + 6 = x². Rearranging and factoring yields:
x² - x - 6 = 0
(x - 3)(x + 2) = 0
Candidates: x = 3 or x = -2
The squared equation permits two candidates. Squaring loses sign information: equal squares do not guarantee equal original quantities. OpenStax's radical-equation guidance, checked October 24, 2026, explains why this operation can introduce extraneous solutions and requires checking the original equation.
Substitute into the original expression
| Candidate | Original left side | Original right side | Disposition |
|---|---|---|---|
| 3 | sqrt(3 + 6) = 3 | 3 | Verified solution |
| -2 | sqrt(-2 + 6) = 2 | -2 | Fails; extraneous candidate |
The only solution is 3. Both candidates satisfy the squared equation, so checking only x² - x - 6 = 0 cannot detect the problem. The candidate -2 also violates the original equation's required x ≥ 0 condition.
This is an authored mathematical example, not an observed ChatGPT answer. If a model returns both roots, ask it to preserve its candidate list and add original-equation substitutions beside the entries rather than silently deleting one.
Request an explanation of equivalence
OpenAI's prompting documentation recommends providing the task and relevant context. For algebra, the domain and meaning of the square-root symbol change what counts as a solution.
Help me review this real-number algebra solution.
Original equation: sqrt(x + 6) = x, using the principal square root.
My work: x + 6 = x²; x² - x - 6 = 0; candidates 3 and -2.
I think both candidates solve the original equation.
Identify the first point where my conclusion needs another check.
State the original domain and sign constraints. Explain whether each
transformation is reversible, or only produces necessary conditions.
Substitute every candidate into the original equation, showing both
sides. Keep candidates separate from verified solutions.
Do not check only the factored equation or use sqrt(a) = ±sqrt(a).
After explaining the issue, give one new radical equation and wait
for my attempt before revealing its solution.
The notation “sqrt(a)” names the principal root. The equation y² = a can have two real roots when a is positive; that does not make the expression sqrt(a) itself both positive and negative.
Check another operation that can lose a solution
For x(x - 4) = 0, dividing immediately by x assumes x is nonzero and loses the valid solution x = 0. Factoring keeps both possibilities visible: x = 0 or x = 4. Substituting each into the original product verifies both.
Squaring can add candidates; division by an expression that may be zero can remove valid ones. Ask what the operation assumes and whether it preserves the full solution set.
For practice, use one-step study prompts after you identify the disputed operation. Keep the original equation and the substitution checks on the same page as your final solution set, so the answer explains why each root belongs there.