4w^2+12w+19=25

Solution for 4w^2+12w+19=25 equation:

Simplifying
4w2 + 12w + 19 = 25

Reorder the terms:
19 + 12w + 4w2 = 25

Solving
19 + 12w + 4w2 = 25

Solving for variable 'w'.

Reorder the terms:
19 + -25 + 12w + 4w2 = 25 + -25

Combine like terms: 19 + -25 = -6
-6 + 12w + 4w2 = 25 + -25

Combine like terms: 25 + -25 = 0
-6 + 12w + 4w2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-3 + 6w + 2w2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-3 + 6w + 2w2)' equal to zero and attempt to solve:

Simplifying
-3 + 6w + 2w2 = 0

Solving
-3 + 6w + 2w2 = 0

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

Divide each side by '2'.
-1.5 + 3w + w2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 3w + 1.5 + w2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 3w + w2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 3w + w2 = 0 + 1.5
3w + w2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
3w + w2 = 1.5

The w term is 3w.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3w + 2.25 + w2 = 1.5 + 2.25

Reorder the terms:
2.25 + 3w + w2 = 1.5 + 2.25

Combine like terms: 1.5 + 2.25 = 3.75
2.25 + 3w + w2 = 3.75

Factor a perfect square on the left side:
(w + 1.5)(w + 1.5) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(w + 1.5) equal to 1.936491673 and -1.936491673.

Subproblem 1
w + 1.5 = 1.936491673

Simplifying
w + 1.5 = 1.936491673

Reorder the terms:
1.5 + w = 1.936491673

Solving
1.5 + w = 1.936491673

Solving for variable 'w'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + w = 1.936491673 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + w = 1.936491673 + -1.5
w = 1.936491673 + -1.5

Combine like terms: 1.936491673 + -1.5 = 0.436491673
w = 0.436491673

Simplifying
w = 0.436491673

Subproblem 2
w + 1.5 = -1.936491673

Simplifying
w + 1.5 = -1.936491673

Reorder the terms:
1.5 + w = -1.936491673

Solving
1.5 + w = -1.936491673

Solving for variable 'w'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + w = -1.936491673 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + w = -1.936491673 + -1.5
w = -1.936491673 + -1.5

Combine like terms: -1.936491673 + -1.5 = -3.436491673
w = -3.436491673

Simplifying
w = -3.436491673

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {0.436491673, -3.436491673}
Solution
w = {0.436491673, -3.436491673}