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Quadric surfaces in standard form

Problem 9.909 · easy

Write \( \displaystyle 2 x^{2} + 72 y^{2} + 18 z^{2} = 72 \) in standard form and identify the surface.

Divide through by 72.
\[ \frac{2 x^{2} + 72 y^{2} + 18 z^{2}}{72} = \frac{x^{2}}{36} + y^{2} + \frac{z^{2}}{4} \]
The standard form.✓ Proved
A ellipsoid: x²/a² + y²/b² + z²/c² = 1.

Answer \( \frac{x^{2}}{36} + y^{2} + \frac{z^{2}}{4} = 1\ (\text{ellipsoid}) \)

Lines: 1 proved, 2 not checked. 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the surface classified from the signs of its coefficients; the standard form is a multiple of the given equation

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (style) — The article 'A' is incorrect before 'ellipsoid'; it should be 'An ellipsoid'. While the math is correct, the grammar error is present in the explanatory sentence.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-11-06 — The article 'A' is incorrect before 'ellipsoid'; it should be 'An ellipsoid'. While the math is correct, the grammar error is present in the explanatory sentence.
gpt-oss:20b: pass 2026-11-06
qwen3.6:27b-mlx: pass 2026-11-05 — The solution correctly divides by 72 to obtain the standard form and correctly identifies the surface as an ellipsoid.
gpt-oss:20b: pass 2026-11-05
qwen3.6:27b-mlx: pass 2026-11-05 — The solution correctly divides by 72 to obtain the standard form and correctly identifies the surface as an ellipsoid. The algebraic steps are sound.
gpt-oss:20b: pass 2026-11-05

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/quadric_surfaces, checked 2026-11-06 with SymPy 1.14.0.