(130+w)(500+w)=3125

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Solution for (130+w)(500+w)=3125 equation: 


Simplifying
(130 + w)(500 + w) = 3125

Multiply (130 + w) * (500 + w)
(130(500 + w) + w(500 + w)) = 3125
((500 * 130 + w * 130) + w(500 + w)) = 3125
((65000 + 130w) + w(500 + w)) = 3125
(65000 + 130w + (500 * w + w * w)) = 3125
(65000 + 130w + (500w + w2)) = 3125

Combine like terms: 130w + 500w = 630w
(65000 + 630w + w2) = 3125

Solving
65000 + 630w + w2 = 3125

Solving for variable 'w'.

Reorder the terms:
65000 + -3125 + 630w + w2 = 3125 + -3125

Combine like terms: 65000 + -3125 = 61875
61875 + 630w + w2 = 3125 + -3125

Combine like terms: 3125 + -3125 = 0
61875 + 630w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-61875' to each side of the equation.
61875 + 630w + -61875 + w2 = 0 + -61875

Reorder the terms:
61875 + -61875 + 630w + w2 = 0 + -61875

Combine like terms: 61875 + -61875 = 0
0 + 630w + w2 = 0 + -61875
630w + w2 = 0 + -61875

Combine like terms: 0 + -61875 = -61875
630w + w2 = -61875

The w term is 630w.  Take half its coefficient (315).
Square it (99225) and add it to both sides.

Add '99225' to each side of the equation.
630w + 99225 + w2 = -61875 + 99225

Reorder the terms:
99225 + 630w + w2 = -61875 + 99225

Combine like terms: -61875 + 99225 = 37350
99225 + 630w + w2 = 37350

Factor a perfect square on the left side:
(w + 315)(w + 315) = 37350

Calculate the square root of the right side: 193.261480901

Break this problem into two subproblems by setting 
(w + 315) equal to 193.261480901 and -193.261480901.


Subproblem 1
w + 315 = 193.261480901

Simplifying
w + 315 = 193.261480901

Reorder the terms:
315 + w = 193.261480901

Solving
315 + w = 193.261480901

Solving for variable 'w'.

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

Add '-315' to each side of the equation.
315 + -315 + w = 193.261480901 + -315

Combine like terms: 315 + -315 = 0
0 + w = 193.261480901 + -315
w = 193.261480901 + -315

Combine like terms: 193.261480901 + -315 = -121.738519099
w = -121.738519099

Simplifying
w = -121.738519099


Subproblem 2
w + 315 = -193.261480901

Simplifying
w + 315 = -193.261480901

Reorder the terms:
315 + w = -193.261480901

Solving
315 + w = -193.261480901

Solving for variable 'w'.

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

Add '-315' to each side of the equation.
315 + -315 + w = -193.261480901 + -315

Combine like terms: 315 + -315 = 0
0 + w = -193.261480901 + -315
w = -193.261480901 + -315

Combine like terms: -193.261480901 + -315 = -508.261480901
w = -508.261480901

Simplifying
w = -508.261480901


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {-121.738519099, -508.261480901}

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