How to Write the Equation of a Sphere in Standard Form: 2x² + 2y² + 2z² + 12x + 24z + 1

To convert the equation of the sphere given by 2x² + 2y² + 2z² + 12x + 24z + 1 = 0 into standard form, we need to follow these steps:

1. Divide the equation by 2: This simplifies the coefficients of the squared terms:
• x² + y² + z² + 6x + 12z + rac{1}{2} = 0
2. Rearrange the equation: Move the constant term to the right side:
• x² + y² + z² + 6x + 12z = -rac{1}{2}
3. Complete the square for x and z: We will complete the square for terms involving x and z.
• For x: x² + 6x can be rewritten as (x + 3)² – 9.
• For z: z² + 12z can be rewritten as (z + 6)² – 36.
4. Substituting back: The equation now looks like:
• (x + 3)² – 9 + (y²) + (z + 6)² – 36 = -rac{1}{2}
5. Simplifying on the left:
• (x + 3)² + y² + (z + 6)² – 45 = -rac{1}{2}
6. Rewrite the equation: Bringing the constants together:
• (x + 3)² + y² + (z + 6)² = rac{89}{2}

Now, the equation is in standard form as:

(x + 3)² + (y – 0)² + (z + 6)² = rac{89}{2}

This represents a sphere with center at (-3, 0, -6) and a radius of √&frac{89}{2}.