52.25=(w)(w+4)

Solution for 52.25=(w)(w+4) equation:

Simplifying
52.25 = (w)(w + 4)

Reorder the terms:
52.25 = w(4 + w)
52.25 = (4 * w + w * w)
52.25 = (4w + w2)

Solving
52.25 = 4w + w2

Solving for variable 'w'.

Reorder the terms:
52.25 + -4w + -1w2 = 4w + -4w + w2 + -1w2

Combine like terms: 4w + -4w = 0
52.25 + -4w + -1w2 = 0 + w2 + -1w2
52.25 + -4w + -1w2 = w2 + -1w2

Combine like terms: w2 + -1w2 = 0
52.25 + -4w + -1w2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-52.25 + 4w + w2 = 0

Move the constant term to the right:

Add '52.25' to each side of the equation.
-52.25 + 4w + 52.25 + w2 = 0 + 52.25

Reorder the terms:
-52.25 + 52.25 + 4w + w2 = 0 + 52.25

Combine like terms: -52.25 + 52.25 = 0.00
0.00 + 4w + w2 = 0 + 52.25
4w + w2 = 0 + 52.25

Combine like terms: 0 + 52.25 = 52.25
4w + w2 = 52.25

The w term is 4w.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4w + 4 + w2 = 52.25 + 4

Reorder the terms:
4 + 4w + w2 = 52.25 + 4

Combine like terms: 52.25 + 4 = 56.25
4 + 4w + w2 = 56.25

Factor a perfect square on the left side:
(w + 2)(w + 2) = 56.25

Calculate the square root of the right side: 7.5

Break this problem into two subproblems by setting 
(w + 2) equal to 7.5 and -7.5.

Subproblem 1
w + 2 = 7.5

Simplifying
w + 2 = 7.5

Reorder the terms:
2 + w = 7.5

Solving
2 + w = 7.5

Solving for variable 'w'.

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

Add '-2' to each side of the equation.
2 + -2 + w = 7.5 + -2

Combine like terms: 2 + -2 = 0
0 + w = 7.5 + -2
w = 7.5 + -2

Combine like terms: 7.5 + -2 = 5.5
w = 5.5

Simplifying
w = 5.5

Subproblem 2
w + 2 = -7.5

Simplifying
w + 2 = -7.5

Reorder the terms:
2 + w = -7.5

Solving
2 + w = -7.5

Solving for variable 'w'.

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

Add '-2' to each side of the equation.
2 + -2 + w = -7.5 + -2

Combine like terms: 2 + -2 = 0
0 + w = -7.5 + -2
w = -7.5 + -2

Combine like terms: -7.5 + -2 = -9.5
w = -9.5

Simplifying
w = -9.5

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {5.5, -9.5}