Quadric surfaces in standard form
Problem 9.739 · easy
Write \( \displaystyle - 4 x^{2} + y^{2} - 16 z = 0 \) in standard form and identify the surface.
- Divide through by 16 and solve for the variable to the first power.
- \[ \frac{- 4 x^{2} + y^{2} - 16 z}{16} = - \frac{x^{2}}{4} + \frac{y^{2}}{16} - z \]The standard form.✓ Proved
- A hyperbolic paraboloid: z to the first power equals a difference of squares.
Answer \( - \frac{x^{2}}{4} + \frac{y^{2}}{16} - z = 0\ (\text{hyperbolic paraboloid}) \)
Lines: 1 proved, 2 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 |
| 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: fail (misleading) — 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 step performed was solving for z when it was merely a simplification.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18gpt-oss:20b: fail (misleading) 2026-10-18 — 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 step performed was solving for z when it was merely a simplification.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly derives the standard form and accurately identifies the surface as a hyperbolic paraboloid.gpt-oss:20b: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The first sentence claims to “solve for the variable to the first power,” yet the step only divides by 16; it does not isolate \(z\). This misleads aqwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly manipulates the equation into the standard form for a hyperbolic paraboloid and correctly identifies the surface.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/quadric_surfaces, checked 2026-10-18 with SymPy 1.14.0.