2n^2-4n=42

Solution for 2n^2-4n=42 equation:

Simplifying
2n2 + -4n = 42

Reorder the terms:
-4n + 2n2 = 42

Solving
-4n + 2n2 = 42

Solving for variable 'n'.

Reorder the terms:
-42 + -4n + 2n2 = 42 + -42

Combine like terms: 42 + -42 = 0
-42 + -4n + 2n2 = 0

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

Ignore the factor 2.
Subproblem 1
Set the factor '(-21 + -2n + n2)' equal to zero and attempt to solve:

Simplifying
-21 + -2n + n2 = 0

Solving
-21 + -2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + -2n + 21 + n2 = 0 + 21

Reorder the terms:
-21 + 21 + -2n + n2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + -2n + n2 = 0 + 21
-2n + n2 = 0 + 21

Combine like terms: 0 + 21 = 21
-2n + n2 = 21

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 = 21 + 1

Reorder the terms:
1 + -2n + n2 = 21 + 1

Combine like terms: 21 + 1 = 22
1 + -2n + n2 = 22

Factor a perfect square on the left side:
(n + -1)(n + -1) = 22

Calculate the square root of the right side: 4.69041576

Break this problem into two subproblems by setting 
(n + -1) equal to 4.69041576 and -4.69041576.

Subproblem 1
n + -1 = 4.69041576

Simplifying
n + -1 = 4.69041576

Reorder the terms:
-1 + n = 4.69041576

Solving
-1 + n = 4.69041576

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 = 4.69041576 + 1

Combine like terms: -1 + 1 = 0
0 + n = 4.69041576 + 1
n = 4.69041576 + 1

Combine like terms: 4.69041576 + 1 = 5.69041576
n = 5.69041576

Simplifying
n = 5.69041576

Subproblem 2
n + -1 = -4.69041576

Simplifying
n + -1 = -4.69041576

Reorder the terms:
-1 + n = -4.69041576

Solving
-1 + n = -4.69041576

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 = -4.69041576 + 1

Combine like terms: -1 + 1 = 0
0 + n = -4.69041576 + 1
n = -4.69041576 + 1

Combine like terms: -4.69041576 + 1 = -3.69041576
n = -3.69041576

Simplifying
n = -3.69041576

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {5.69041576, -3.69041576}
Solution
n = {5.69041576, -3.69041576}