n^2+14n=-22

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Solution for n^2+14n=-22 equation:


Simplifying
n2 + 14n = -22

Reorder the terms:
14n + n2 = -22

Solving
14n + n2 = -22

Solving for variable 'n'.

Reorder the terms:
22 + 14n + n2 = -22 + 22

Combine like terms: -22 + 22 = 0
22 + 14n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-22' to each side of the equation.
22 + 14n + -22 + n2 = 0 + -22

Reorder the terms:
22 + -22 + 14n + n2 = 0 + -22

Combine like terms: 22 + -22 = 0
0 + 14n + n2 = 0 + -22
14n + n2 = 0 + -22

Combine like terms: 0 + -22 = -22
14n + n2 = -22

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 = -22 + 49

Reorder the terms:
49 + 14n + n2 = -22 + 49

Combine like terms: -22 + 49 = 27
49 + 14n + n2 = 27

Factor a perfect square on the left side:
(n + 7)(n + 7) = 27

Calculate the square root of the right side: 5.196152423

Break this problem into two subproblems by setting 
(n + 7) equal to 5.196152423 and -5.196152423.

Subproblem 1

n + 7 = 5.196152423 Simplifying n + 7 = 5.196152423 Reorder the terms: 7 + n = 5.196152423 Solving 7 + n = 5.196152423 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 = 5.196152423 + -7 Combine like terms: 7 + -7 = 0 0 + n = 5.196152423 + -7 n = 5.196152423 + -7 Combine like terms: 5.196152423 + -7 = -1.803847577 n = -1.803847577 Simplifying n = -1.803847577

Subproblem 2

n + 7 = -5.196152423 Simplifying n + 7 = -5.196152423 Reorder the terms: 7 + n = -5.196152423 Solving 7 + n = -5.196152423 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 = -5.196152423 + -7 Combine like terms: 7 + -7 = 0 0 + n = -5.196152423 + -7 n = -5.196152423 + -7 Combine like terms: -5.196152423 + -7 = -12.196152423 n = -12.196152423 Simplifying n = -12.196152423

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-1.803847577, -12.196152423}

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