(5n^2-n+6)+(8n^2-11)=0

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Solution for (5n^2-n+6)+(8n^2-11)=0 equation:


Simplifying
(5n2 + -1n + 6) + (8n2 + -11) = 0

Reorder the terms:
(6 + -1n + 5n2) + (8n2 + -11) = 0

Remove parenthesis around (6 + -1n + 5n2)
6 + -1n + 5n2 + (8n2 + -11) = 0

Reorder the terms:
6 + -1n + 5n2 + (-11 + 8n2) = 0

Remove parenthesis around (-11 + 8n2)
6 + -1n + 5n2 + -11 + 8n2 = 0

Reorder the terms:
6 + -11 + -1n + 5n2 + 8n2 = 0

Combine like terms: 6 + -11 = -5
-5 + -1n + 5n2 + 8n2 = 0

Combine like terms: 5n2 + 8n2 = 13n2
-5 + -1n + 13n2 = 0

Solving
-5 + -1n + 13n2 = 0

Solving for variable 'n'.

Begin completing the square.  Divide all terms by
13 the coefficient of the squared term: 

Divide each side by '13'.
-0.3846153846 + -0.07692307692n + n2 = 0

Move the constant term to the right:

Add '0.3846153846' to each side of the equation.
-0.3846153846 + -0.07692307692n + 0.3846153846 + n2 = 0 + 0.3846153846

Reorder the terms:
-0.3846153846 + 0.3846153846 + -0.07692307692n + n2 = 0 + 0.3846153846

Combine like terms: -0.3846153846 + 0.3846153846 = 0.0000000000
0.0000000000 + -0.07692307692n + n2 = 0 + 0.3846153846
-0.07692307692n + n2 = 0 + 0.3846153846

Combine like terms: 0 + 0.3846153846 = 0.3846153846
-0.07692307692n + n2 = 0.3846153846

The n term is -0.07692307692n.  Take half its coefficient (-0.03846153846).
Square it (0.001479289941) and add it to both sides.

Add '0.001479289941' to each side of the equation.
-0.07692307692n + 0.001479289941 + n2 = 0.3846153846 + 0.001479289941

Reorder the terms:
0.001479289941 + -0.07692307692n + n2 = 0.3846153846 + 0.001479289941

Combine like terms: 0.3846153846 + 0.001479289941 = 0.386094674541
0.001479289941 + -0.07692307692n + n2 = 0.386094674541

Factor a perfect square on the left side:
(n + -0.03846153846)(n + -0.03846153846) = 0.386094674541

Calculate the square root of the right side: 0.62136517

Break this problem into two subproblems by setting 
(n + -0.03846153846) equal to 0.62136517 and -0.62136517.

Subproblem 1

n + -0.03846153846 = 0.62136517 Simplifying n + -0.03846153846 = 0.62136517 Reorder the terms: -0.03846153846 + n = 0.62136517 Solving -0.03846153846 + n = 0.62136517 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.03846153846' to each side of the equation. -0.03846153846 + 0.03846153846 + n = 0.62136517 + 0.03846153846 Combine like terms: -0.03846153846 + 0.03846153846 = 0.00000000000 0.00000000000 + n = 0.62136517 + 0.03846153846 n = 0.62136517 + 0.03846153846 Combine like terms: 0.62136517 + 0.03846153846 = 0.65982670846 n = 0.65982670846 Simplifying n = 0.65982670846

Subproblem 2

n + -0.03846153846 = -0.62136517 Simplifying n + -0.03846153846 = -0.62136517 Reorder the terms: -0.03846153846 + n = -0.62136517 Solving -0.03846153846 + n = -0.62136517 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.03846153846' to each side of the equation. -0.03846153846 + 0.03846153846 + n = -0.62136517 + 0.03846153846 Combine like terms: -0.03846153846 + 0.03846153846 = 0.00000000000 0.00000000000 + n = -0.62136517 + 0.03846153846 n = -0.62136517 + 0.03846153846 Combine like terms: -0.62136517 + 0.03846153846 = -0.58290363154 n = -0.58290363154 Simplifying n = -0.58290363154

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.65982670846, -0.58290363154}

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