4w^2-31w+12=0

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Solution for 4w^2-31w+12=0 equation: 


Simplifying
4w2 + -31w + 12 = 0

Reorder the terms:
12 + -31w + 4w2 = 0

Solving
12 + -31w + 4w2 = 0

Solving for variable 'w'.

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

Divide each side by '4'.
3 + -7.75w + w2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -7.75w + -3 + w2 = 0 + -3

Reorder the terms:
3 + -3 + -7.75w + w2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -7.75w + w2 = 0 + -3
-7.75w + w2 = 0 + -3

Combine like terms: 0 + -3 = -3
-7.75w + w2 = -3

The w term is -7.75w.  Take half its coefficient (-3.875).
Square it (15.015625) and add it to both sides.

Add '15.015625' to each side of the equation.
-7.75w + 15.015625 + w2 = -3 + 15.015625

Reorder the terms:
15.015625 + -7.75w + w2 = -3 + 15.015625

Combine like terms: -3 + 15.015625 = 12.015625
15.015625 + -7.75w + w2 = 12.015625

Factor a perfect square on the left side:
(w + -3.875)(w + -3.875) = 12.015625

Calculate the square root of the right side: 3.466356156

Break this problem into two subproblems by setting 
(w + -3.875) equal to 3.466356156 and -3.466356156.


Subproblem 1
w + -3.875 = 3.466356156

Simplifying
w + -3.875 = 3.466356156

Reorder the terms:
-3.875 + w = 3.466356156

Solving
-3.875 + w = 3.466356156

Solving for variable 'w'.

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

Add '3.875' to each side of the equation.
-3.875 + 3.875 + w = 3.466356156 + 3.875

Combine like terms: -3.875 + 3.875 = 0.000
0.000 + w = 3.466356156 + 3.875
w = 3.466356156 + 3.875

Combine like terms: 3.466356156 + 3.875 = 7.341356156
w = 7.341356156

Simplifying
w = 7.341356156


Subproblem 2
w + -3.875 = -3.466356156

Simplifying
w + -3.875 = -3.466356156

Reorder the terms:
-3.875 + w = -3.466356156

Solving
-3.875 + w = -3.466356156

Solving for variable 'w'.

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

Add '3.875' to each side of the equation.
-3.875 + 3.875 + w = -3.466356156 + 3.875

Combine like terms: -3.875 + 3.875 = 0.000
0.000 + w = -3.466356156 + 3.875
w = -3.466356156 + 3.875

Combine like terms: -3.466356156 + 3.875 = 0.408643844
w = 0.408643844

Simplifying
w = 0.408643844


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {7.341356156, 0.408643844}

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