Home›Calculus 3›Quadric surfaces in standard form›Problem 9.908

Quadric surfaces in standard form

Problem 9.908 · easy

Write \( \displaystyle 4 x^{2} + 16 y^{2} - 16 z = 0 \) in standard form and identify the surface.

Divide through by 16 and solve for the variable to the first power.
Reviewed
\[ \frac{4 x^{2} + 16 y^{2} - 16 z}{16} = \frac{x^{2}}{4} + y^{2} - z \]
The standard form.✓ Proved
A elliptic paraboloid: z (or another variable) to the first power equals a sum of squares.
Reviewed

Answer \( \frac{x^{2}}{4} + y^{2} - z = 0\ (\text{elliptic paraboloid}) \)

Lines: 1 proved, 2 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
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the standard form and the surface type. The reasoning is sound and the steps are clear.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly identifies the standard form and the surface type. The reasoning is sound and the steps are clear.
gpt-oss:20b: pass 2026-11-06
qwen3.6:27b-mlx: pass 2026-11-05 — The solution correctly identifies the algebraic steps to reach standard form and accurately identifies the surface type based on the resulting equation.
gpt-oss:20b: fail (misleading) 2026-11-05 — The first sentence claims to “solve for the variable to the first power,” but the solution only divides by 16 and does not isolate z. This could mislead a student into thinking the equation is already solved for z when it is not.
qwen3.6:27b-mlx: fail (style) 2026-11-05 — The phrase 'solve for the variable to the first power' is imprecise; one does not 'solve' for z in this context, but rather isolates it to match the standard form z = ... . Additionally, 'A elliptic' should be 'An elliptic'.
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.