Question

Write the equation of the sphere in standard form. x^2 + y^2 + z^2 + 2x − 2y − 2z = 22

Answer 1

Given:
x^2 + y^2 + z^2 + 2x − 2y − 2z = 22

group the variables
x^2+2x + y^2-2y +z^2-2z = 22

then complete the squares
x^2+2x+1 + y^2-2y+1 + z^2-2z+1 = 22 +1 +1 +1

factor and simplify
(x+1)^2 + (y-1)^2 + (z-1)^2 = 5^2

Interpretation:
The centre of the sphere is at (-1,1,1) with a radius of 5