2n^2+12n+11=0

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Solution for 2n^2+12n+11=0 equation:


Simplifying
2n2 + 12n + 11 = 0

Reorder the terms:
11 + 12n + 2n2 = 0

Solving
11 + 12n + 2n2 = 0

Solving for variable 'n'.

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

Divide each side by '2'.
5.5 + 6n + n2 = 0

Move the constant term to the right:

Add '-5.5' to each side of the equation.
5.5 + 6n + -5.5 + n2 = 0 + -5.5

Reorder the terms:
5.5 + -5.5 + 6n + n2 = 0 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + 6n + n2 = 0 + -5.5
6n + n2 = 0 + -5.5

Combine like terms: 0 + -5.5 = -5.5
6n + n2 = -5.5

The n term is 6n.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6n + 9 + n2 = -5.5 + 9

Reorder the terms:
9 + 6n + n2 = -5.5 + 9

Combine like terms: -5.5 + 9 = 3.5
9 + 6n + n2 = 3.5

Factor a perfect square on the left side:
(n + 3)(n + 3) = 3.5

Calculate the square root of the right side: 1.870828693

Break this problem into two subproblems by setting 
(n + 3) equal to 1.870828693 and -1.870828693.

Subproblem 1

n + 3 = 1.870828693 Simplifying n + 3 = 1.870828693 Reorder the terms: 3 + n = 1.870828693 Solving 3 + n = 1.870828693 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = 1.870828693 + -3 Combine like terms: 3 + -3 = 0 0 + n = 1.870828693 + -3 n = 1.870828693 + -3 Combine like terms: 1.870828693 + -3 = -1.129171307 n = -1.129171307 Simplifying n = -1.129171307

Subproblem 2

n + 3 = -1.870828693 Simplifying n + 3 = -1.870828693 Reorder the terms: 3 + n = -1.870828693 Solving 3 + n = -1.870828693 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = -1.870828693 + -3 Combine like terms: 3 + -3 = 0 0 + n = -1.870828693 + -3 n = -1.870828693 + -3 Combine like terms: -1.870828693 + -3 = -4.870828693 n = -4.870828693 Simplifying n = -4.870828693

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-1.129171307, -4.870828693}

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