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Equations of spheres

Problem 9.612 · easy

Find the center and radius of the sphere \( \displaystyle x^{2} + y^{2} + z^{2} - 8 z + 15 = 0 \).

Complete the square in x, y and z.
Reviewed
\[ x^{2} + y^{2} + z^{2} - 8 z + 15 = x^{2} + y^{2} + \left(z - 4\right)^{2} - 1 \]
So (x − 0)² + (y − 0)² + (z − 4)² = 1.✓ Proved

Answer \( \text{center } (0, 0, 4),\ \text{radius } 1 \)

Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	points at the claimed radius from the claimed center satisfy the equation

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly completes the square to identify the center and radius. The algebraic manipulation is verified, and the final result matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly completes the square to identify the center and radius. The algebraic manipulation is verified, and the final result matches the stated answer.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly completes the square to identify the center and radius. The algebraic manipulation is verified, and the final result matches the stated answer.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The algebraic manipulation in step 2 is incorrect. Expanding (z - 4)^2 yields z^2 - 8z + 16, so z^2 - 8z + 15 = (z - 4)^2 - 1. The solution incorrectly claims the expression equals (x^2 + y^2 + (z - 4)^2) - 1, which implies z^2 - 8z + 15 = z^2 - 8z + 16 - 1 = z^2 - 8z + 15, but the grouping suggests the constant term was handled incorrectly relative to the standard form derivation. Specifically, x^2 + y^2 + z^2 - 8z + 15 = 0 becomes x^2 + y^2 + (z-4)^2 - 16 + 15 = 0, so x^2 + y^2 + (z-4)^2 = 1. The intermediate equation shown is technically an identity if interpreted as LHS = RHS, but the notation is confusing and the 'unchecked' tag hides the fact that the transition to the standard form is not clearly justified by the text. More importantly, the prompt asks to review sentences. The sentence 'Complete the square...' is fine. The equation is marked 'unchecked' but the prompt says 'each line... is either an EQUATION... or a SENTENCE'. If the equation is unchecked, I should check the logic. The logic is: x^2 + y^2 + z^2 - 8z + 15 = 0. Add 1 to both sides? No. z^2 - 8z = (z-4)^2 - 16. So x^2 + y^2 + (z-4)^2 - 16 + 15 = 0 => x^2 + y^2 + (z-4)^2 = 1. The solution writes: x**2 + y**2 + z**2 - 8*z + 15 = (x**2 + y**2 + (z - 4)**2) - 1. Let's check RHS: x^2 + y^2 + z^2 - 8z + 16 - 1 = x^2 + y^2 + z^2 - 8z + 15. This is correct. Then it says '-- So (x − 0)² + (y − 0)² + (z − 4)² = 1.' This implies setting the LHS to 0. If LHS = 0, then RHS = 0, so (x-0)^2 + ... = 1. The logic holds. Why did I think it was an error? Let me re-read carefully. 'x**2 + y**2 + z**2 - 8*z + 15 = (x**2 + y**2 + (z - 4)**2) - 1'. This is an identity. The problem states the equation is equal to 0. So 0 = (x-0)^2 + (y-0)^2 + (z-4)^2 - 1. Thus (x-0)^2 + (y-0)^2 + (z-4)^2 = 1. Center (0,0,4), radius 1. The solution is correct. The sentence 'Complete the square...' is correct. The equation is correct. The conclusion is correct. Wait, is there a subtle error? No. It seems correct. Let me double check the 'unchecked' part. The prompt says 'each line... is either an EQUATION... or a SENTENCE'. If it's an equation, the CAS checked it. If it's unchecked, maybe it's a sentence? No, it's labeled [equation, unchecked]. This is contradictory to the prompt instructions which say 'EQUATION, which a computer algebra system has already checked'. Perhaps 'unchecked' means I should check it? Or perhaps it's a typo in the prompt format provided to me? Assuming I must check the validity. The validity is correct. So verdict pass.
gpt-oss:20b: pass 2026-10-14

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/sphere_equation, checked 2026-10-17 with SymPy 1.14.0.