2w^2+2w=42

Solution for 2w^2+2w=42 equation:

Simplifying
2w2 + 2w = 42

Reorder the terms:
2w + 2w2 = 42

Solving
2w + 2w2 = 42

Solving for variable 'w'.

Reorder the terms:
-42 + 2w + 2w2 = 42 + -42

Combine like terms: 42 + -42 = 0
-42 + 2w + 2w2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-21 + w + w2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-21 + w + w2)' equal to zero and attempt to solve:

Simplifying
-21 + w + w2 = 0

Solving
-21 + w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + w + 21 + w2 = 0 + 21

Reorder the terms:
-21 + 21 + w + w2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + w + w2 = 0 + 21
w + w2 = 0 + 21

Combine like terms: 0 + 21 = 21
w + w2 = 21

The w term is w.  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.
w + 0.25 + w2 = 21 + 0.25

Reorder the terms:
0.25 + w + w2 = 21 + 0.25

Combine like terms: 21 + 0.25 = 21.25
0.25 + w + w2 = 21.25

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

Calculate the square root of the right side: 4.609772229

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

Subproblem 1
w + 0.5 = 4.609772229

Simplifying
w + 0.5 = 4.609772229

Reorder the terms:
0.5 + w = 4.609772229

Solving
0.5 + w = 4.609772229

Solving for variable 'w'.

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

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

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

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

Simplifying
w = 4.109772229

Subproblem 2
w + 0.5 = -4.609772229

Simplifying
w + 0.5 = -4.609772229

Reorder the terms:
0.5 + w = -4.609772229

Solving
0.5 + w = -4.609772229

Solving for variable 'w'.

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

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

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

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

Simplifying
w = -5.109772229

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