n^2=-14n-37

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Solution for n^2=-14n-37 equation:


Simplifying
n2 = -14n + -37

Reorder the terms:
n2 = -37 + -14n

Solving
n2 = -37 + -14n

Solving for variable 'n'.

Reorder the terms:
37 + 14n + n2 = -37 + -14n + 37 + 14n

Reorder the terms:
37 + 14n + n2 = -37 + 37 + -14n + 14n

Combine like terms: -37 + 37 = 0
37 + 14n + n2 = 0 + -14n + 14n
37 + 14n + n2 = -14n + 14n

Combine like terms: -14n + 14n = 0
37 + 14n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-37' to each side of the equation.
37 + 14n + -37 + n2 = 0 + -37

Reorder the terms:
37 + -37 + 14n + n2 = 0 + -37

Combine like terms: 37 + -37 = 0
0 + 14n + n2 = 0 + -37
14n + n2 = 0 + -37

Combine like terms: 0 + -37 = -37
14n + n2 = -37

The n term is 14n.  Take half its coefficient (7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
14n + 49 + n2 = -37 + 49

Reorder the terms:
49 + 14n + n2 = -37 + 49

Combine like terms: -37 + 49 = 12
49 + 14n + n2 = 12

Factor a perfect square on the left side:
(n + 7)(n + 7) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(n + 7) equal to 3.464101615 and -3.464101615.

Subproblem 1

n + 7 = 3.464101615 Simplifying n + 7 = 3.464101615 Reorder the terms: 7 + n = 3.464101615 Solving 7 + n = 3.464101615 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + n = 3.464101615 + -7 Combine like terms: 7 + -7 = 0 0 + n = 3.464101615 + -7 n = 3.464101615 + -7 Combine like terms: 3.464101615 + -7 = -3.535898385 n = -3.535898385 Simplifying n = -3.535898385

Subproblem 2

n + 7 = -3.464101615 Simplifying n + 7 = -3.464101615 Reorder the terms: 7 + n = -3.464101615 Solving 7 + n = -3.464101615 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + n = -3.464101615 + -7 Combine like terms: 7 + -7 = 0 0 + n = -3.464101615 + -7 n = -3.464101615 + -7 Combine like terms: -3.464101615 + -7 = -10.464101615 n = -10.464101615 Simplifying n = -10.464101615

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-3.535898385, -10.464101615}

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