8n-2(2n-7)(2n+6)=11n

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Solution for 8n-2(2n-7)(2n+6)=11n equation:


Simplifying
8n + -2(2n + -7)(2n + 6) = 11n

Reorder the terms:
8n + -2(-7 + 2n)(2n + 6) = 11n

Reorder the terms:
8n + -2(-7 + 2n)(6 + 2n) = 11n

Multiply (-7 + 2n) * (6 + 2n)
8n + -2(-7(6 + 2n) + 2n * (6 + 2n)) = 11n
8n + -2((6 * -7 + 2n * -7) + 2n * (6 + 2n)) = 11n
8n + -2((-42 + -14n) + 2n * (6 + 2n)) = 11n
8n + -2(-42 + -14n + (6 * 2n + 2n * 2n)) = 11n
8n + -2(-42 + -14n + (12n + 4n2)) = 11n

Combine like terms: -14n + 12n = -2n
8n + -2(-42 + -2n + 4n2) = 11n
8n + (-42 * -2 + -2n * -2 + 4n2 * -2) = 11n
8n + (84 + 4n + -8n2) = 11n

Reorder the terms:
84 + 8n + 4n + -8n2 = 11n

Combine like terms: 8n + 4n = 12n
84 + 12n + -8n2 = 11n

Solving
84 + 12n + -8n2 = 11n

Solving for variable 'n'.

Reorder the terms:
84 + 12n + -11n + -8n2 = 11n + -11n

Combine like terms: 12n + -11n = 1n
84 + 1n + -8n2 = 11n + -11n

Combine like terms: 11n + -11n = 0
84 + 1n + -8n2 = 0

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
-10.5 + -0.125n + n2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + -0.125n + 10.5 + n2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + -0.125n + n2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + -0.125n + n2 = 0 + 10.5
-0.125n + n2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
-0.125n + n2 = 10.5

The n term is -0.125n.  Take half its coefficient (-0.0625).
Square it (0.00390625) and add it to both sides.

Add '0.00390625' to each side of the equation.
-0.125n + 0.00390625 + n2 = 10.5 + 0.00390625

Reorder the terms:
0.00390625 + -0.125n + n2 = 10.5 + 0.00390625

Combine like terms: 10.5 + 0.00390625 = 10.50390625
0.00390625 + -0.125n + n2 = 10.50390625

Factor a perfect square on the left side:
(n + -0.0625)(n + -0.0625) = 10.50390625

Calculate the square root of the right side: 3.240973041

Break this problem into two subproblems by setting 
(n + -0.0625) equal to 3.240973041 and -3.240973041.

Subproblem 1

n + -0.0625 = 3.240973041 Simplifying n + -0.0625 = 3.240973041 Reorder the terms: -0.0625 + n = 3.240973041 Solving -0.0625 + n = 3.240973041 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + n = 3.240973041 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + n = 3.240973041 + 0.0625 n = 3.240973041 + 0.0625 Combine like terms: 3.240973041 + 0.0625 = 3.303473041 n = 3.303473041 Simplifying n = 3.303473041

Subproblem 2

n + -0.0625 = -3.240973041 Simplifying n + -0.0625 = -3.240973041 Reorder the terms: -0.0625 + n = -3.240973041 Solving -0.0625 + n = -3.240973041 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + n = -3.240973041 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + n = -3.240973041 + 0.0625 n = -3.240973041 + 0.0625 Combine like terms: -3.240973041 + 0.0625 = -3.178473041 n = -3.178473041 Simplifying n = -3.178473041

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.303473041, -3.178473041}

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