2w^2-17w+31-w^2-10w-25=0

Solution for 2w^2-17w+31-w^2-10w-25=0 equation:

Simplifying
2w2 + -17w + 31 + -1w2 + -10w + -25 = 0

Reorder the terms:
31 + -25 + -17w + -10w + 2w2 + -1w2 = 0

Combine like terms: 31 + -25 = 6
6 + -17w + -10w + 2w2 + -1w2 = 0

Combine like terms: -17w + -10w = -27w
6 + -27w + 2w2 + -1w2 = 0

Combine like terms: 2w2 + -1w2 = 1w2
6 + -27w + 1w2 = 0

Solving
6 + -27w + 1w2 = 0

Solving for variable 'w'.

Begin completing the square.

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -27w + -6 + w2 = 0 + -6

Reorder the terms:
6 + -6 + -27w + w2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -27w + w2 = 0 + -6
-27w + w2 = 0 + -6

Combine like terms: 0 + -6 = -6
-27w + w2 = -6

The w term is -27w.  Take half its coefficient (-13.5).
Square it (182.25) and add it to both sides.

Add '182.25' to each side of the equation.
-27w + 182.25 + w2 = -6 + 182.25

Reorder the terms:
182.25 + -27w + w2 = -6 + 182.25

Combine like terms: -6 + 182.25 = 176.25
182.25 + -27w + w2 = 176.25

Factor a perfect square on the left side:
(w + -13.5)(w + -13.5) = 176.25

Calculate the square root of the right side: 13.275918047

Break this problem into two subproblems by setting 
(w + -13.5) equal to 13.275918047 and -13.275918047.

Subproblem 1
w + -13.5 = 13.275918047

Simplifying
w + -13.5 = 13.275918047

Reorder the terms:
-13.5 + w = 13.275918047

Solving
-13.5 + w = 13.275918047

Solving for variable 'w'.

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

Add '13.5' to each side of the equation.
-13.5 + 13.5 + w = 13.275918047 + 13.5

Combine like terms: -13.5 + 13.5 = 0.0
0.0 + w = 13.275918047 + 13.5
w = 13.275918047 + 13.5

Combine like terms: 13.275918047 + 13.5 = 26.775918047
w = 26.775918047

Simplifying
w = 26.775918047

Subproblem 2
w + -13.5 = -13.275918047

Simplifying
w + -13.5 = -13.275918047

Reorder the terms:
-13.5 + w = -13.275918047

Solving
-13.5 + w = -13.275918047

Solving for variable 'w'.

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

Add '13.5' to each side of the equation.
-13.5 + 13.5 + w = -13.275918047 + 13.5

Combine like terms: -13.5 + 13.5 = 0.0
0.0 + w = -13.275918047 + 13.5
w = -13.275918047 + 13.5

Combine like terms: -13.275918047 + 13.5 = 0.224081953
w = 0.224081953

Simplifying
w = 0.224081953

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {26.775918047, 0.224081953}