4w^2-20w+22=0

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Solution for 4w^2-20w+22=0 equation: 


Simplifying
4w2 + -20w + 22 = 0

Reorder the terms:
22 + -20w + 4w2 = 0

Solving
22 + -20w + 4w2 = 0

Solving for variable 'w'.

Factor out the Greatest Common Factor (GCF), '2'.
2(11 + -10w + 2w2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(11 + -10w + 2w2)' equal to zero and attempt to solve:

Simplifying
11 + -10w + 2w2 = 0

Solving
11 + -10w + 2w2 = 0

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

Divide each side by '2'.
5.5 + -5w + w2 = 0

Move the constant term to the right:

Add '-5.5' to each side of the equation.
5.5 + -5w + -5.5 + w2 = 0 + -5.5

Reorder the terms:
5.5 + -5.5 + -5w + w2 = 0 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + -5w + w2 = 0 + -5.5
-5w + w2 = 0 + -5.5

Combine like terms: 0 + -5.5 = -5.5
-5w + w2 = -5.5

The w term is -5w.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5w + 6.25 + w2 = -5.5 + 6.25

Reorder the terms:
6.25 + -5w + w2 = -5.5 + 6.25

Combine like terms: -5.5 + 6.25 = 0.75
6.25 + -5w + w2 = 0.75

Factor a perfect square on the left side:
(w + -2.5)(w + -2.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(w + -2.5) equal to 0.866025404 and -0.866025404.


Subproblem 1
w + -2.5 = 0.866025404

Simplifying
w + -2.5 = 0.866025404

Reorder the terms:
-2.5 + w = 0.866025404

Solving
-2.5 + w = 0.866025404

Solving for variable 'w'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + w = 0.866025404 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + w = 0.866025404 + 2.5
w = 0.866025404 + 2.5

Combine like terms: 0.866025404 + 2.5 = 3.366025404
w = 3.366025404

Simplifying
w = 3.366025404


Subproblem 2
w + -2.5 = -0.866025404

Simplifying
w + -2.5 = -0.866025404

Reorder the terms:
-2.5 + w = -0.866025404

Solving
-2.5 + w = -0.866025404

Solving for variable 'w'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + w = -0.866025404 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + w = -0.866025404 + 2.5
w = -0.866025404 + 2.5

Combine like terms: -0.866025404 + 2.5 = 1.633974596
w = 1.633974596

Simplifying
w = 1.633974596


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {3.366025404, 1.633974596}
Solution
w = {3.366025404, 1.633974596}

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