n*n+8*n=11

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Solution for n*n+8*n=11 equation:


Simplifying
n * n + 8n = 11

Multiply n * n
n2 + 8n = 11

Reorder the terms:
8n + n2 = 11

Solving
8n + n2 = 11

Solving for variable 'n'.

Reorder the terms:
-11 + 8n + n2 = 11 + -11

Combine like terms: 11 + -11 = 0
-11 + 8n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + 8n + 11 + n2 = 0 + 11

Reorder the terms:
-11 + 11 + 8n + n2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + 8n + n2 = 0 + 11
8n + n2 = 0 + 11

Combine like terms: 0 + 11 = 11
8n + n2 = 11

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 = 11 + 16

Reorder the terms:
16 + 8n + n2 = 11 + 16

Combine like terms: 11 + 16 = 27
16 + 8n + n2 = 27

Factor a perfect square on the left side:
(n + 4)(n + 4) = 27

Calculate the square root of the right side: 5.196152423

Break this problem into two subproblems by setting 
(n + 4) equal to 5.196152423 and -5.196152423.

Subproblem 1

n + 4 = 5.196152423 Simplifying n + 4 = 5.196152423 Reorder the terms: 4 + n = 5.196152423 Solving 4 + n = 5.196152423 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 = 5.196152423 + -4 Combine like terms: 4 + -4 = 0 0 + n = 5.196152423 + -4 n = 5.196152423 + -4 Combine like terms: 5.196152423 + -4 = 1.196152423 n = 1.196152423 Simplifying n = 1.196152423

Subproblem 2

n + 4 = -5.196152423 Simplifying n + 4 = -5.196152423 Reorder the terms: 4 + n = -5.196152423 Solving 4 + n = -5.196152423 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 = -5.196152423 + -4 Combine like terms: 4 + -4 = 0 0 + n = -5.196152423 + -4 n = -5.196152423 + -4 Combine like terms: -5.196152423 + -4 = -9.196152423 n = -9.196152423 Simplifying n = -9.196152423

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.196152423, -9.196152423}

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