4w^2-21w+22=0

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Solution for 4w^2-21w+22=0 equation:


Simplifying
4w2 + -21w + 22 = 0

Reorder the terms:
22 + -21w + 4w2 = 0

Solving
22 + -21w + 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'.
5.5 + -5.25w + w2 = 0

Move the constant term to the right:

Add '-5.5' to each side of the equation.
5.5 + -5.25w + -5.5 + w2 = 0 + -5.5

Reorder the terms:
5.5 + -5.5 + -5.25w + w2 = 0 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + -5.25w + w2 = 0 + -5.5
-5.25w + w2 = 0 + -5.5

Combine like terms: 0 + -5.5 = -5.5
-5.25w + w2 = -5.5

The w term is -5.25w.  Take half its coefficient (-2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
-5.25w + 6.890625 + w2 = -5.5 + 6.890625

Reorder the terms:
6.890625 + -5.25w + w2 = -5.5 + 6.890625

Combine like terms: -5.5 + 6.890625 = 1.390625
6.890625 + -5.25w + w2 = 1.390625

Factor a perfect square on the left side:
(w + -2.625)(w + -2.625) = 1.390625

Calculate the square root of the right side: 1.179247642

Break this problem into two subproblems by setting 
(w + -2.625) equal to 1.179247642 and -1.179247642.

Subproblem 1

w + -2.625 = 1.179247642 Simplifying w + -2.625 = 1.179247642 Reorder the terms: -2.625 + w = 1.179247642 Solving -2.625 + w = 1.179247642 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + w = 1.179247642 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + w = 1.179247642 + 2.625 w = 1.179247642 + 2.625 Combine like terms: 1.179247642 + 2.625 = 3.804247642 w = 3.804247642 Simplifying w = 3.804247642

Subproblem 2

w + -2.625 = -1.179247642 Simplifying w + -2.625 = -1.179247642 Reorder the terms: -2.625 + w = -1.179247642 Solving -2.625 + w = -1.179247642 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + w = -1.179247642 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + w = -1.179247642 + 2.625 w = -1.179247642 + 2.625 Combine like terms: -1.179247642 + 2.625 = 1.445752358 w = 1.445752358 Simplifying w = 1.445752358

Solution

The solution to the problem is based on the solutions from the subproblems. w = {3.804247642, 1.445752358}

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