Complete the square to write the following equation in the standard form for a sphere. $$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 y + 2 z + 16 = 0$$
A) $( x + 2 ) ^ { 2 } + ( y + 2 ) ^ { 2 } + ( z + 1 ) ^ { 2 } = 25$
B) $( x - 2 ) ^ { 2 } + ( y + 2 ) ^ { 2 } + ( z + 1 ) ^ { 2 } = 25$
C) $( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z + 1 ) ^ { 2 } = 25$
D) $( x - 2 ) ^ { 2 } + ( y + 2 ) ^ { 2 } + ( z + 1 ) ^ { 2 } = 5$
E) $( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 25$

Question

Quizplus · Accepted Answer

To complete the square for each variable, we group the terms by variable and adjust the constant term accordingly. For $x^2 - 4x$, we add and subtract $(\frac{-4}{2})^2 = 4$. For $y^2 + 4y$, we add and subtract $(\frac{4}{2})^2 = 4$. For $z^2 + 2z$, we add and subtract $(\frac{2}{2})^2 = 1$. Adding these inside the equation gives us:$$x^2 - 4x + 4 + y^2 + 4y + 4 + z^2 + 2z + 1 = 0 + 4 + 4 + 1 - 16$$$$= (x - 2)^2 + (y + 2)^2 + (z + 1)^2 = 25$$This matches option B.

Complete the Square to Write the Following Equation in the Standard