n^2+16n-89=0

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Solution for n^2+16n-89=0 equation:


Simplifying
n2 + 16n + -89 = 0

Reorder the terms:
-89 + 16n + n2 = 0

Solving
-89 + 16n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '89' to each side of the equation.
-89 + 16n + 89 + n2 = 0 + 89

Reorder the terms:
-89 + 89 + 16n + n2 = 0 + 89

Combine like terms: -89 + 89 = 0
0 + 16n + n2 = 0 + 89
16n + n2 = 0 + 89

Combine like terms: 0 + 89 = 89
16n + n2 = 89

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 = 89 + 64

Reorder the terms:
64 + 16n + n2 = 89 + 64

Combine like terms: 89 + 64 = 153
64 + 16n + n2 = 153

Factor a perfect square on the left side:
(n + 8)(n + 8) = 153

Calculate the square root of the right side: 12.369316877

Break this problem into two subproblems by setting 
(n + 8) equal to 12.369316877 and -12.369316877.

Subproblem 1

n + 8 = 12.369316877 Simplifying n + 8 = 12.369316877 Reorder the terms: 8 + n = 12.369316877 Solving 8 + n = 12.369316877 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 = 12.369316877 + -8 Combine like terms: 8 + -8 = 0 0 + n = 12.369316877 + -8 n = 12.369316877 + -8 Combine like terms: 12.369316877 + -8 = 4.369316877 n = 4.369316877 Simplifying n = 4.369316877

Subproblem 2

n + 8 = -12.369316877 Simplifying n + 8 = -12.369316877 Reorder the terms: 8 + n = -12.369316877 Solving 8 + n = -12.369316877 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 = -12.369316877 + -8 Combine like terms: 8 + -8 = 0 0 + n = -12.369316877 + -8 n = -12.369316877 + -8 Combine like terms: -12.369316877 + -8 = -20.369316877 n = -20.369316877 Simplifying n = -20.369316877

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.369316877, -20.369316877}

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