2n^2-4n-14=0

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Solution for 2n^2-4n-14=0 equation: 


Simplifying
2n2 + -4n + -14 = 0

Reorder the terms:
-14 + -4n + 2n2 = 0

Solving
-14 + -4n + 2n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-7 + -2n + n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-7 + -2n + n2)' equal to zero and attempt to solve:

Simplifying
-7 + -2n + n2 = 0

Solving
-7 + -2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -2n + 7 + n2 = 0 + 7

Reorder the terms:
-7 + 7 + -2n + n2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -2n + n2 = 0 + 7
-2n + n2 = 0 + 7

Combine like terms: 0 + 7 = 7
-2n + n2 = 7

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 7 + 1

Reorder the terms:
1 + -2n + n2 = 7 + 1

Combine like terms: 7 + 1 = 8
1 + -2n + n2 = 8

Factor a perfect square on the left side:
(n + -1)(n + -1) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(n + -1) equal to 2.828427125 and -2.828427125.


Subproblem 1
n + -1 = 2.828427125

Simplifying
n + -1 = 2.828427125

Reorder the terms:
-1 + n = 2.828427125

Solving
-1 + n = 2.828427125

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '1' to each side of the equation.
-1 + 1 + n = 2.828427125 + 1

Combine like terms: -1 + 1 = 0
0 + n = 2.828427125 + 1
n = 2.828427125 + 1

Combine like terms: 2.828427125 + 1 = 3.828427125
n = 3.828427125

Simplifying
n = 3.828427125


Subproblem 2
n + -1 = -2.828427125

Simplifying
n + -1 = -2.828427125

Reorder the terms:
-1 + n = -2.828427125

Solving
-1 + n = -2.828427125

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '1' to each side of the equation.
-1 + 1 + n = -2.828427125 + 1

Combine like terms: -1 + 1 = 0
0 + n = -2.828427125 + 1
n = -2.828427125 + 1

Combine like terms: -2.828427125 + 1 = -1.828427125
n = -1.828427125

Simplifying
n = -1.828427125


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {3.828427125, -1.828427125}
Solution
n = {3.828427125, -1.828427125}

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