31w+5w^2=30

Solution for 31w+5w^2=30 equation:

Simplifying
31w + 5w2 = 30

Solving
31w + 5w2 = 30

Solving for variable 'w'.

Reorder the terms:
-30 + 31w + 5w2 = 30 + -30

Combine like terms: 30 + -30 = 0
-30 + 31w + 5w2 = 0

Begin completing the square.  Divide all terms by
5 the coefficient of the squared term: 

Divide each side by '5'.
-6 + 6.2w + w2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 6.2w + 6 + w2 = 0 + 6

Reorder the terms:
-6 + 6 + 6.2w + w2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 6.2w + w2 = 0 + 6
6.2w + w2 = 0 + 6

Combine like terms: 0 + 6 = 6
6.2w + w2 = 6

The w term is 6.2w.  Take half its coefficient (3.1).
Square it (9.61) and add it to both sides.

Add '9.61' to each side of the equation.
6.2w + 9.61 + w2 = 6 + 9.61

Reorder the terms:
9.61 + 6.2w + w2 = 6 + 9.61

Combine like terms: 6 + 9.61 = 15.61
9.61 + 6.2w + w2 = 15.61

Factor a perfect square on the left side:
(w + 3.1)(w + 3.1) = 15.61

Calculate the square root of the right side: 3.950949253

Break this problem into two subproblems by setting 
(w + 3.1) equal to 3.950949253 and -3.950949253.

Subproblem 1
w + 3.1 = 3.950949253

Simplifying
w + 3.1 = 3.950949253

Reorder the terms:
3.1 + w = 3.950949253

Solving
3.1 + w = 3.950949253

Solving for variable 'w'.

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

Add '-3.1' to each side of the equation.
3.1 + -3.1 + w = 3.950949253 + -3.1

Combine like terms: 3.1 + -3.1 = 0.0
0.0 + w = 3.950949253 + -3.1
w = 3.950949253 + -3.1

Combine like terms: 3.950949253 + -3.1 = 0.850949253
w = 0.850949253

Simplifying
w = 0.850949253

Subproblem 2
w + 3.1 = -3.950949253

Simplifying
w + 3.1 = -3.950949253

Reorder the terms:
3.1 + w = -3.950949253

Solving
3.1 + w = -3.950949253

Solving for variable 'w'.

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

Add '-3.1' to each side of the equation.
3.1 + -3.1 + w = -3.950949253 + -3.1

Combine like terms: 3.1 + -3.1 = 0.0
0.0 + w = -3.950949253 + -3.1
w = -3.950949253 + -3.1

Combine like terms: -3.950949253 + -3.1 = -7.050949253
w = -7.050949253

Simplifying
w = -7.050949253

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {0.850949253, -7.050949253}