(2w+10)(2w+4)=w(w+5)+190

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Solution for (2w+10)(2w+4)=w(w+5)+190 equation: 


Simplifying
(2w + 10)(2w + 4) = w(w + 5) + 190

Reorder the terms:
(10 + 2w)(2w + 4) = w(w + 5) + 190

Reorder the terms:
(10 + 2w)(4 + 2w) = w(w + 5) + 190

Multiply (10 + 2w) * (4 + 2w)
(10(4 + 2w) + 2w * (4 + 2w)) = w(w + 5) + 190
((4 * 10 + 2w * 10) + 2w * (4 + 2w)) = w(w + 5) + 190
((40 + 20w) + 2w * (4 + 2w)) = w(w + 5) + 190
(40 + 20w + (4 * 2w + 2w * 2w)) = w(w + 5) + 190
(40 + 20w + (8w + 4w2)) = w(w + 5) + 190

Combine like terms: 20w + 8w = 28w
(40 + 28w + 4w2) = w(w + 5) + 190

Reorder the terms:
40 + 28w + 4w2 = w(5 + w) + 190
40 + 28w + 4w2 = (5 * w + w * w) + 190
40 + 28w + 4w2 = (5w + w2) + 190

Reorder the terms:
40 + 28w + 4w2 = 190 + 5w + w2

Solving
40 + 28w + 4w2 = 190 + 5w + w2

Solving for variable 'w'.

Reorder the terms:
40 + -190 + 28w + -5w + 4w2 + -1w2 = 190 + 5w + w2 + -190 + -5w + -1w2

Combine like terms: 40 + -190 = -150
-150 + 28w + -5w + 4w2 + -1w2 = 190 + 5w + w2 + -190 + -5w + -1w2

Combine like terms: 28w + -5w = 23w
-150 + 23w + 4w2 + -1w2 = 190 + 5w + w2 + -190 + -5w + -1w2

Combine like terms: 4w2 + -1w2 = 3w2
-150 + 23w + 3w2 = 190 + 5w + w2 + -190 + -5w + -1w2

Reorder the terms:
-150 + 23w + 3w2 = 190 + -190 + 5w + -5w + w2 + -1w2

Combine like terms: 190 + -190 = 0
-150 + 23w + 3w2 = 0 + 5w + -5w + w2 + -1w2
-150 + 23w + 3w2 = 5w + -5w + w2 + -1w2

Combine like terms: 5w + -5w = 0
-150 + 23w + 3w2 = 0 + w2 + -1w2
-150 + 23w + 3w2 = w2 + -1w2

Combine like terms: w2 + -1w2 = 0
-150 + 23w + 3w2 = 0

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

Divide each side by '3'.
-50 + 7.666666667w + w2 = 0

Move the constant term to the right:

Add '50' to each side of the equation.
-50 + 7.666666667w + 50 + w2 = 0 + 50

Reorder the terms:
-50 + 50 + 7.666666667w + w2 = 0 + 50

Combine like terms: -50 + 50 = 0
0 + 7.666666667w + w2 = 0 + 50
7.666666667w + w2 = 0 + 50

Combine like terms: 0 + 50 = 50
7.666666667w + w2 = 50

The w term is 7.666666667w.  Take half its coefficient (3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
7.666666667w + 14.69444445 + w2 = 50 + 14.69444445

Reorder the terms:
14.69444445 + 7.666666667w + w2 = 50 + 14.69444445

Combine like terms: 50 + 14.69444445 = 64.69444445
14.69444445 + 7.666666667w + w2 = 64.69444445

Factor a perfect square on the left side:
(w + 3.833333334)(w + 3.833333334) = 64.69444445

Calculate the square root of the right side: 8.043285675

Break this problem into two subproblems by setting 
(w + 3.833333334) equal to 8.043285675 and -8.043285675.


Subproblem 1
w + 3.833333334 = 8.043285675

Simplifying
w + 3.833333334 = 8.043285675

Reorder the terms:
3.833333334 + w = 8.043285675

Solving
3.833333334 + w = 8.043285675

Solving for variable 'w'.

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

Add '-3.833333334' to each side of the equation.
3.833333334 + -3.833333334 + w = 8.043285675 + -3.833333334

Combine like terms: 3.833333334 + -3.833333334 = 0.000000000
0.000000000 + w = 8.043285675 + -3.833333334
w = 8.043285675 + -3.833333334

Combine like terms: 8.043285675 + -3.833333334 = 4.209952341
w = 4.209952341

Simplifying
w = 4.209952341


Subproblem 2
w + 3.833333334 = -8.043285675

Simplifying
w + 3.833333334 = -8.043285675

Reorder the terms:
3.833333334 + w = -8.043285675

Solving
3.833333334 + w = -8.043285675

Solving for variable 'w'.

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

Add '-3.833333334' to each side of the equation.
3.833333334 + -3.833333334 + w = -8.043285675 + -3.833333334

Combine like terms: 3.833333334 + -3.833333334 = 0.000000000
0.000000000 + w = -8.043285675 + -3.833333334
w = -8.043285675 + -3.833333334

Combine like terms: -8.043285675 + -3.833333334 = -11.876619009
w = -11.876619009

Simplifying
w = -11.876619009


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {4.209952341, -11.876619009}

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