n^2+12+4n+11=76

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Solution for n^2+12+4n+11=76 equation:


Simplifying
n2 + 12 + 4n + 11 = 76

Reorder the terms:
12 + 11 + 4n + n2 = 76

Combine like terms: 12 + 11 = 23
23 + 4n + n2 = 76

Solving
23 + 4n + n2 = 76

Solving for variable 'n'.

Reorder the terms:
23 + -76 + 4n + n2 = 76 + -76

Combine like terms: 23 + -76 = -53
-53 + 4n + n2 = 76 + -76

Combine like terms: 76 + -76 = 0
-53 + 4n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '53' to each side of the equation.
-53 + 4n + 53 + n2 = 0 + 53

Reorder the terms:
-53 + 53 + 4n + n2 = 0 + 53

Combine like terms: -53 + 53 = 0
0 + 4n + n2 = 0 + 53
4n + n2 = 0 + 53

Combine like terms: 0 + 53 = 53
4n + n2 = 53

The n term is 4n.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4n + 4 + n2 = 53 + 4

Reorder the terms:
4 + 4n + n2 = 53 + 4

Combine like terms: 53 + 4 = 57
4 + 4n + n2 = 57

Factor a perfect square on the left side:
(n + 2)(n + 2) = 57

Calculate the square root of the right side: 7.549834435

Break this problem into two subproblems by setting 
(n + 2) equal to 7.549834435 and -7.549834435.

Subproblem 1

n + 2 = 7.549834435 Simplifying n + 2 = 7.549834435 Reorder the terms: 2 + n = 7.549834435 Solving 2 + n = 7.549834435 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 7.549834435 + -2 Combine like terms: 2 + -2 = 0 0 + n = 7.549834435 + -2 n = 7.549834435 + -2 Combine like terms: 7.549834435 + -2 = 5.549834435 n = 5.549834435 Simplifying n = 5.549834435

Subproblem 2

n + 2 = -7.549834435 Simplifying n + 2 = -7.549834435 Reorder the terms: 2 + n = -7.549834435 Solving 2 + n = -7.549834435 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -7.549834435 + -2 Combine like terms: 2 + -2 = 0 0 + n = -7.549834435 + -2 n = -7.549834435 + -2 Combine like terms: -7.549834435 + -2 = -9.549834435 n = -9.549834435 Simplifying n = -9.549834435

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.549834435, -9.549834435}

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