Quadric surfaces in standard form
Problem 9.732 · easy
Write \( \displaystyle - 3 x^{2} + 75 y^{2} - 75 z = 0 \) in standard form and identify the surface.
- Divide through by 75 and solve for the variable to the first power.Reviewed
- \[ \frac{- 3 x^{2} + 75 y^{2} - 75 z}{75} = - \frac{x^{2}}{25} + y^{2} - z \]The standard form.✓ Proved
- A hyperbolic paraboloid: z to the first power equals a difference of squares.Reviewed
Answer \( - \frac{x^{2}}{25} + y^{2} - z = 0\ (\text{hyperbolic paraboloid}) \)
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 — The first sentence is vague—‘solve for the variable to the first power’ is unnecessary and could confuse a student, but the mathematical steps are correct.qwen3.6:27b-mlx: pass — The solution correctly derives the standard form and accurately identifies the surface as a hyperbolic paraboloid based on the structure of the equation.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly derives the standard form and accurately identifies the surface as a hyperbolic paraboloid based on the structure of the equation.gpt-oss:20b: pass 2026-10-18 — The first sentence is vague—‘solve for the variable to the first power’ is unnecessary and could confuse a student, but the mathematical steps are correct.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly derives the standard form and accurately identifies the surface based on the equation structure.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the standard form and the surface type. The reasoning is sound and concise.gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence claims to solve for the variable to the first power, but it only divides by 75; it does not isolate z or otherwise solve for a linear variable. This misleads a student into thinking a further step was performed.
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.