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Equations of spheres

Problem 9.709 · easy

Find the center and radius of the sphere \( \displaystyle x^{2} - 6 x + y^{2} + 4 y + z^{2} + 2 z + 5 = 0 \).

Complete the square in x, y and z.
\[ x^{2} - 6 x + y^{2} + 4 y + z^{2} + 2 z + 5 = \left(x - 3\right)^{2} + \left(y + 2\right)^{2} + \left(z + 1\right)^{2} - 9 \]
So (x − 3)² + (y − -2)² + (z − -1)² = 9.✓ Proved

Answer \( \text{center } (3, -2, -1),\ \text{radius } 3 \)

Lines: 1 proved, 1 not checked. 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	Not checked	—	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: fail (error) — The algebraic completion of the square is incorrect. The constant term should be -13 (since -9 + 4 + 1 + 5 = -1), not -9. Consequently, the radius squared is 13, not 9.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The algebraic completion of the square is incorrect. The constant term should be -13 (since -9 + 4 + 1 + 5 = -1), not -9. Consequently, the radius squared is 13, not 9.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly completes the square for all variables, derives the standard form of the sphere equation, and identifies the center and radius accurately.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The algebraic completion of the square is incorrect. The sum of the subtracted constants is 9 + 4 + 1 = 14, not 9. Consequently, the radius squared should be 14 - 5 = 9, but the intermediate step shown claims the constant term is -9, which is mathematically false.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.