n^2+9n+11=0

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Solution for n^2+9n+11=0 equation:


Simplifying
n2 + 9n + 11 = 0

Reorder the terms:
11 + 9n + n2 = 0

Solving
11 + 9n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-11' to each side of the equation.
11 + 9n + -11 + n2 = 0 + -11

Reorder the terms:
11 + -11 + 9n + n2 = 0 + -11

Combine like terms: 11 + -11 = 0
0 + 9n + n2 = 0 + -11
9n + n2 = 0 + -11

Combine like terms: 0 + -11 = -11
9n + n2 = -11

The n term is 9n.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9n + 20.25 + n2 = -11 + 20.25

Reorder the terms:
20.25 + 9n + n2 = -11 + 20.25

Combine like terms: -11 + 20.25 = 9.25
20.25 + 9n + n2 = 9.25

Factor a perfect square on the left side:
(n + 4.5)(n + 4.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(n + 4.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

n + 4.5 = 3.041381265 Simplifying n + 4.5 = 3.041381265 Reorder the terms: 4.5 + n = 3.041381265 Solving 4.5 + n = 3.041381265 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = 3.041381265 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = 3.041381265 + -4.5 n = 3.041381265 + -4.5 Combine like terms: 3.041381265 + -4.5 = -1.458618735 n = -1.458618735 Simplifying n = -1.458618735

Subproblem 2

n + 4.5 = -3.041381265 Simplifying n + 4.5 = -3.041381265 Reorder the terms: 4.5 + n = -3.041381265 Solving 4.5 + n = -3.041381265 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = -3.041381265 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = -3.041381265 + -4.5 n = -3.041381265 + -4.5 Combine like terms: -3.041381265 + -4.5 = -7.541381265 n = -7.541381265 Simplifying n = -7.541381265

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-1.458618735, -7.541381265}

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