2n^2-12n-22=0

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


Simplifying
2n2 + -12n + -22 = 0

Reorder the terms:
-22 + -12n + 2n2 = 0

Solving
-22 + -12n + 2n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-11 + -6n + n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-11 + -6n + n2)' equal to zero and attempt to solve:

Simplifying
-11 + -6n + n2 = 0

Solving
-11 + -6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + -6n + 11 + n2 = 0 + 11

Reorder the terms:
-11 + 11 + -6n + n2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + -6n + n2 = 0 + 11
-6n + n2 = 0 + 11

Combine like terms: 0 + 11 = 11
-6n + n2 = 11

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 = 11 + 9

Reorder the terms:
9 + -6n + n2 = 11 + 9

Combine like terms: 11 + 9 = 20
9 + -6n + n2 = 20

Factor a perfect square on the left side:
(n + -3)(n + -3) = 20

Calculate the square root of the right side: 4.472135955

Break this problem into two subproblems by setting 
(n + -3) equal to 4.472135955 and -4.472135955.


Subproblem 1
n + -3 = 4.472135955

Simplifying
n + -3 = 4.472135955

Reorder the terms:
-3 + n = 4.472135955

Solving
-3 + n = 4.472135955

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 = 4.472135955 + 3

Combine like terms: -3 + 3 = 0
0 + n = 4.472135955 + 3
n = 4.472135955 + 3

Combine like terms: 4.472135955 + 3 = 7.472135955
n = 7.472135955

Simplifying
n = 7.472135955


Subproblem 2
n + -3 = -4.472135955

Simplifying
n + -3 = -4.472135955

Reorder the terms:
-3 + n = -4.472135955

Solving
-3 + n = -4.472135955

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 = -4.472135955 + 3

Combine like terms: -3 + 3 = 0
0 + n = -4.472135955 + 3
n = -4.472135955 + 3

Combine like terms: -4.472135955 + 3 = -1.472135955
n = -1.472135955

Simplifying
n = -1.472135955


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {7.472135955, -1.472135955}
Solution
n = {7.472135955, -1.472135955}

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