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Equations of spheres

Problem 9.799 · easy

Find the center and radius of the sphere \( \displaystyle x^{2} - 6 x + y^{2} + 10 y + z^{2} - 6 z + 34 = 0 \).

Complete the square in x, y and z.
\[ x^{2} - 6 x + y^{2} + 10 y + z^{2} - 6 z + 34 = \left(x - 3\right)^{2} + \left(y + 5\right)^{2} + \left(z - 3\right)^{2} - 9 \]
So (x − 3)² + (y − -5)² + (z − 3)² = 9.✓ Proved

Answer \( \text{center } (3, -5, 3),\ \text{radius } 3 \)

Lines: 1 proved, 1 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
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: fail (error) — The algebraic manipulation in line 2 is incorrect. Completing the square for the given expression results in (x - 3)^2 + (y + 5)^2 + (z - 3)^2 - 49, not -9. Consequently, the radius squared is 49 (radius 7), not 9.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The algebraic manipulation in line 2 is incorrect. Completing the square for the given expression results in (x - 3)^2 + (y + 5)^2 + (z - 3)^2 - 49, not -9. Consequently, the radius squared is 49 (radius 7), not 9.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The algebraic manipulation in line 2 is incorrect. Completing the square for x, y, and z terms yields a constant of -9 - 25 - 9 = -43, not -9. Consequently, the equation becomes (x-3)^2 + (y+5)^2 + (z-3)^2 = 9, which implies a radius of 3, but the original equation actually has no real solutions (radius squared is -9) or the constant term was copied incorrectly in the derivation. Specifically, 34 - 43 = -9, so the sum of squares equals -9, meaning no real sphere exists. The stated answer claims a real sphere with radius 3, which is mathematically impossible for the given equation.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The algebraic manipulation in step 2 is incorrect. Completing the square for x, y, and z yields constants -9, -25, and -9, summing to -43. The equation should be (x-3)^2 + (y+5)^2 + (z-3)^2 = 9, but the intermediate step shown claims the sum of squares minus 9 equals the original expression, ignoring the -25 and -9 from y and z. Specifically, (x-3)^2 + (y+5)^2 + (z-3)^2 = x^2 - 6x + 9 + y^2 + 10y + 25 + z^2 - 6z + 9 = (original expression - 34) + 43 = original expression + 9. Thus, original expression = 0 implies sum of squares = 9. The provided line says sum of squares - 9 = original expression, which implies sum of squares = original expression + 9. This is actually correct for the final result, but the text says '- 9' instead of '- 43' or similar to account for all terms. Let's re-read carefully: 'x**2 - 6*x + y**2 + 10*y + z**2 - 6*z + 34 = ((x - 3)**2 + (y + 5)**2 + (z - 3)**2) - 9'. LHS is the original expression. RHS is SumOfSquares - 9. If LHS=0, then SumOfSquares=9. This is the correct final equation. However, is the equality LHS = RHS true? LHS = x^2 - 6x + y^2 + 10y + z^2 - 6z + 34. RHS = (x^2 - 6x + 9) + (y^2 + 10y + 25) + (z^2 - 6z + 9) - 9 = x^2 - 6x + y^2 + 10y + z^2 - 6z + 34 - 9 + 9? No. 9+25+9 = 43. 43-9 = 34. So RHS = x^2 - 6x + y^2 + 10y + z^2 - 6z + 34. The equality holds. The step is algebraically correct. Wait, let me double check. (x-3)^2 = x^2-6x+9. (y+5)^2 = y^2+10y+25. (z-3)^2 = z^2-6z+9. Sum = x^2-6x+y^2+10y+z^2-6z + 43. The equation says LHS = Sum - 9. Sum - 9 = x^2-6x+y^2+10y+z^2-6z + 34. This matches LHS. So the equation is correct. The conclusion 'So (x-3)^2 + ... = 9' follows from setting LHS=0. So 0 = Sum - 9 => Sum = 9. This is correct. Why did I think it was wrong? I misread the constant adjustment. The solution is actually correct. Let me re-verify the problem statement. Center (3, -5, 3), radius 3. Radius squared is 9. Correct. The solution is sound.
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/sphere_equation, checked 2026-10-19 with SymPy 1.14.0.