n^2+12n+22=0

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Solution for n^2+12n+22=0 equation:


Simplifying
n2 + 12n + 22 = 0

Reorder the terms:
22 + 12n + n2 = 0

Solving
22 + 12n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-22' to each side of the equation.
22 + 12n + -22 + n2 = 0 + -22

Reorder the terms:
22 + -22 + 12n + n2 = 0 + -22

Combine like terms: 22 + -22 = 0
0 + 12n + n2 = 0 + -22
12n + n2 = 0 + -22

Combine like terms: 0 + -22 = -22
12n + n2 = -22

The n term is 12n.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12n + 36 + n2 = -22 + 36

Reorder the terms:
36 + 12n + n2 = -22 + 36

Combine like terms: -22 + 36 = 14
36 + 12n + n2 = 14

Factor a perfect square on the left side:
(n + 6)(n + 6) = 14

Calculate the square root of the right side: 3.741657387

Break this problem into two subproblems by setting 
(n + 6) equal to 3.741657387 and -3.741657387.

Subproblem 1

n + 6 = 3.741657387 Simplifying n + 6 = 3.741657387 Reorder the terms: 6 + n = 3.741657387 Solving 6 + n = 3.741657387 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + n = 3.741657387 + -6 Combine like terms: 6 + -6 = 0 0 + n = 3.741657387 + -6 n = 3.741657387 + -6 Combine like terms: 3.741657387 + -6 = -2.258342613 n = -2.258342613 Simplifying n = -2.258342613

Subproblem 2

n + 6 = -3.741657387 Simplifying n + 6 = -3.741657387 Reorder the terms: 6 + n = -3.741657387 Solving 6 + n = -3.741657387 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + n = -3.741657387 + -6 Combine like terms: 6 + -6 = 0 0 + n = -3.741657387 + -6 n = -3.741657387 + -6 Combine like terms: -3.741657387 + -6 = -9.741657387 n = -9.741657387 Simplifying n = -9.741657387

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-2.258342613, -9.741657387}

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