n^2+5=-8n

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Solution for n^2+5=-8n equation:


Simplifying
n2 + 5 = -8n

Reorder the terms:
5 + n2 = -8n

Solving
5 + n2 = -8n

Solving for variable 'n'.

Reorder the terms:
5 + 8n + n2 = -8n + 8n

Combine like terms: -8n + 8n = 0
5 + 8n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 8n + -5 + n2 = 0 + -5

Reorder the terms:
5 + -5 + 8n + n2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 8n + n2 = 0 + -5
8n + n2 = 0 + -5

Combine like terms: 0 + -5 = -5
8n + n2 = -5

The n term is 8n.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8n + 16 + n2 = -5 + 16

Reorder the terms:
16 + 8n + n2 = -5 + 16

Combine like terms: -5 + 16 = 11
16 + 8n + n2 = 11

Factor a perfect square on the left side:
(n + 4)(n + 4) = 11

Calculate the square root of the right side: 3.31662479

Break this problem into two subproblems by setting 
(n + 4) equal to 3.31662479 and -3.31662479.

Subproblem 1

n + 4 = 3.31662479 Simplifying n + 4 = 3.31662479 Reorder the terms: 4 + n = 3.31662479 Solving 4 + n = 3.31662479 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + n = 3.31662479 + -4 Combine like terms: 4 + -4 = 0 0 + n = 3.31662479 + -4 n = 3.31662479 + -4 Combine like terms: 3.31662479 + -4 = -0.68337521 n = -0.68337521 Simplifying n = -0.68337521

Subproblem 2

n + 4 = -3.31662479 Simplifying n + 4 = -3.31662479 Reorder the terms: 4 + n = -3.31662479 Solving 4 + n = -3.31662479 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + n = -3.31662479 + -4 Combine like terms: 4 + -4 = 0 0 + n = -3.31662479 + -4 n = -3.31662479 + -4 Combine like terms: -3.31662479 + -4 = -7.31662479 n = -7.31662479 Simplifying n = -7.31662479

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.68337521, -7.31662479}

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