Sphere Equations and Completing the Square Quiz

What is the center-radius form of a sphere equation?

• (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2

• x^2 + y^2 + z^2 + ax + by + cz + d = 0

• x^2 + y^2 + z^2 = r^2

• (x + a)^2 + (y + b)^2 + (z + c)^2 = r^2

What is the purpose of completing the square in the context of sphere equations?

• To find the volume of the sphere

• To convert the equation to the general form

• To determine the center and radius from the general form

• To simplify the equation for easier graphing

In completing the square, what do you add to the expression x^2 + 10x to make it a perfect square?

• 10

• 100

• 25

• 5

When completing the square for y^2 - 8y, what value is added to complete the square?

• 16

• 8

• 64

• 4

In the example with x^2 + y^2 + z^2 + 2x + 4y - 6z - 2 = 0, what is the radius of the sphere?

• 8

• 4

• 16

• 2

For the equation x^2 + y^2 + z^2 - x + 5z = 0, what is the center of the sphere?

• (-1/2, 0, -5/2)

• (1/2, 0, 5/2)

• (-1/2, 0, 5/2)

• (1/2, 0, -5/2)

In the equation 3x^2 + 3y^2 + 3z^2 + 6x + y = 0, what is the first step to simplify the equation?

• Subtract 3 from each term

• Divide each term by 3

• Multiply each term by 3

• Add 3 to each term

What is the center of the sphere for the equation x^2 + y^2 + z^2 + 2x + 1/3y = 0 after completing the square?

• (1, -1/6, 0)

• (-1, -1/6, 0)

• (1, 1/6, 0)

• (-1, 1/6, 0)

What is the radius of the sphere for the equation x^2 + y^2 + z^2 + 2x + 1/3y = 0 after completing the square?

• √37/2

• √37/6

• √37/3

• √37/36

In the context of sphere equations, why is it important for the coefficients of x^2, y^2, and z^2 to be the same?

• To make the equation easier to graph

• To ensure the equation represents a circle

• To simplify the calculation of the volume

• To ensure the equation represents a sphere