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Equations of spheres

Problem 9.806 · easy

Find the center and radius of the sphere \( \displaystyle x^{2} - 4 x + y^{2} - 2 y + z^{2} + 8 z + 5 = 0 \).

Complete the square in x, y and z.
Reviewed
\[ x^{2} - 4 x + y^{2} - 2 y + z^{2} + 8 z + 5 = \left(x - 2\right)^{2} + \left(y - 1\right)^{2} + \left(z + 4\right)^{2} - 16 \]
So (x − 2)² + (y − 1)² + (z − -4)² = 16.✓ Proved

Answer \( \text{center } (2, 1, -4),\ \text{radius } 4 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 for all variables and derives the standard form of the sphere equation. The center and radius are correctly identified from the resulting equation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly completes the square for all variables and derives the standard form of the sphere equation. The center and radius are correctly identified from the resulting equation.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly completes the square for all variables and derives the standard form of the sphere equation. The center and radius are correctly identified from the resulting equation.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly completes the square for all variables and derives the standard form of the sphere equation, leading to the correct center and radius.
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 2026-10-19 with SymPy 1.14.0.