n^2=32-16n

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Solution for n^2=32-16n equation:


Simplifying
n2 = 32 + -16n

Solving
n2 = 32 + -16n

Solving for variable 'n'.

Reorder the terms:
-32 + 16n + n2 = 32 + -16n + -32 + 16n

Reorder the terms:
-32 + 16n + n2 = 32 + -32 + -16n + 16n

Combine like terms: 32 + -32 = 0
-32 + 16n + n2 = 0 + -16n + 16n
-32 + 16n + n2 = -16n + 16n

Combine like terms: -16n + 16n = 0
-32 + 16n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '32' to each side of the equation.
-32 + 16n + 32 + n2 = 0 + 32

Reorder the terms:
-32 + 32 + 16n + n2 = 0 + 32

Combine like terms: -32 + 32 = 0
0 + 16n + n2 = 0 + 32
16n + n2 = 0 + 32

Combine like terms: 0 + 32 = 32
16n + n2 = 32

The n term is 16n.  Take half its coefficient (8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
16n + 64 + n2 = 32 + 64

Reorder the terms:
64 + 16n + n2 = 32 + 64

Combine like terms: 32 + 64 = 96
64 + 16n + n2 = 96

Factor a perfect square on the left side:
(n + 8)(n + 8) = 96

Calculate the square root of the right side: 9.797958971

Break this problem into two subproblems by setting 
(n + 8) equal to 9.797958971 and -9.797958971.

Subproblem 1

n + 8 = 9.797958971 Simplifying n + 8 = 9.797958971 Reorder the terms: 8 + n = 9.797958971 Solving 8 + n = 9.797958971 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + n = 9.797958971 + -8 Combine like terms: 8 + -8 = 0 0 + n = 9.797958971 + -8 n = 9.797958971 + -8 Combine like terms: 9.797958971 + -8 = 1.797958971 n = 1.797958971 Simplifying n = 1.797958971

Subproblem 2

n + 8 = -9.797958971 Simplifying n + 8 = -9.797958971 Reorder the terms: 8 + n = -9.797958971 Solving 8 + n = -9.797958971 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + n = -9.797958971 + -8 Combine like terms: 8 + -8 = 0 0 + n = -9.797958971 + -8 n = -9.797958971 + -8 Combine like terms: -9.797958971 + -8 = -17.797958971 n = -17.797958971 Simplifying n = -17.797958971

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.797958971, -17.797958971}

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