2n^2+2n-42=0

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Solution for 2n^2+2n-42=0 equation: 


Simplifying
2n2 + 2n + -42 = 0

Reorder the terms:
-42 + 2n + 2n2 = 0

Solving
-42 + 2n + 2n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-21 + n + n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-21 + n + n2)' equal to zero and attempt to solve:

Simplifying
-21 + n + n2 = 0

Solving
-21 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + n + 21 + n2 = 0 + 21

Reorder the terms:
-21 + 21 + n + n2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + n + n2 = 0 + 21
n + n2 = 0 + 21

Combine like terms: 0 + 21 = 21
n + n2 = 21

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
n + 0.25 + n2 = 21 + 0.25

Reorder the terms:
0.25 + n + n2 = 21 + 0.25

Combine like terms: 21 + 0.25 = 21.25
0.25 + n + n2 = 21.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(n + 0.5) equal to 4.609772229 and -4.609772229.


Subproblem 1
n + 0.5 = 4.609772229

Simplifying
n + 0.5 = 4.609772229

Reorder the terms:
0.5 + n = 4.609772229

Solving
0.5 + n = 4.609772229

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = 4.609772229 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 4.609772229 + -0.5
n = 4.609772229 + -0.5

Combine like terms: 4.609772229 + -0.5 = 4.109772229
n = 4.109772229

Simplifying
n = 4.109772229


Subproblem 2
n + 0.5 = -4.609772229

Simplifying
n + 0.5 = -4.609772229

Reorder the terms:
0.5 + n = -4.609772229

Solving
0.5 + n = -4.609772229

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = -4.609772229 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -4.609772229 + -0.5
n = -4.609772229 + -0.5

Combine like terms: -4.609772229 + -0.5 = -5.109772229
n = -5.109772229

Simplifying
n = -5.109772229


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.109772229, -5.109772229}
Solution
n = {4.109772229, -5.109772229}

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