Final answer:
The corrected equation x² + 6x + y² = -8 defines a circle in the Cartesian plane with a center at (-3,0) and a radius of √17, once the typo is addressed and the square is completed.
Step-by-step explanation:
The student has presented the equation x² + y² + 972 + 6x = -8, which appears to contain a typo or irrelevant parts. If we ignore the '+972' which seems out of place for typical geometric equations, the equation simplifies to x² + 6x + y² = -8. This resembles the equation for a circle in the Cartesian plane, but it needs to be completed to a perfect square to clearly identify the center and radius of the circle.
Using the quadratic formula, we can find the roots for equations of the form ax² + bx + c = 0. However, to identify the surface defined by the equation x² + y² + 6x = -8, we should complete the square for the x-terms:
- Move the constant term to the other side: x² + 6x = -y² + 8
- Add (b/2)², where b is the coefficient of x, to both sides to complete the square: x² + 6x + 9 = -y² + 8 + 9
- Simplify to get the equation of the circle: (x + 3)² + y² = 17
Thus, the proper equation without the typo defines a circle with a center at (-3,0) and a radius of √17.