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Equations of spheres

Problem 9.620 · easy

Find the center and radius of the sphere \( \displaystyle x^{2} + 4 x + y^{2} + 8 y + z^{2} - 2 z + 20 = 0 \).

Complete the square in x, y and z.
Reviewed
\[ x^{2} + 4 x + y^{2} + 8 y + z^{2} - 2 z + 20 = \left(x + 2\right)^{2} + \left(y + 4\right)^{2} + \left(z - 1\right)^{2} - 1 \]
So (x − -2)² + (y − -4)² + (z − 1)² = 1.✓ Proved

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

✓ 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 equation of the sphere, leading to the correct center and radius.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly completes the square for all variables and derives the standard equation of the sphere, leading to the correct center and radius.
gpt-oss:20b: pass 2026-10-17
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. The identified center and radius are correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — 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 2026-10-14

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.