180=w(30+w)

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Solution for 180=w(30+w) equation:


Simplifying
180 = w(30 + w)
180 = (30 * w + w * w)
180 = (30w + w2)

Solving
180 = 30w + w2

Solving for variable 'w'.

Reorder the terms:
180 + -30w + -1w2 = 30w + -30w + w2 + -1w2

Combine like terms: 30w + -30w = 0
180 + -30w + -1w2 = 0 + w2 + -1w2
180 + -30w + -1w2 = w2 + -1w2

Combine like terms: w2 + -1w2 = 0
180 + -30w + -1w2 = 0

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

Divide each side by '-1'.
-180 + 30w + w2 = 0

Move the constant term to the right:

Add '180' to each side of the equation.
-180 + 30w + 180 + w2 = 0 + 180

Reorder the terms:
-180 + 180 + 30w + w2 = 0 + 180

Combine like terms: -180 + 180 = 0
0 + 30w + w2 = 0 + 180
30w + w2 = 0 + 180

Combine like terms: 0 + 180 = 180
30w + w2 = 180

The w term is 30w.  Take half its coefficient (15).
Square it (225) and add it to both sides.

Add '225' to each side of the equation.
30w + 225 + w2 = 180 + 225

Reorder the terms:
225 + 30w + w2 = 180 + 225

Combine like terms: 180 + 225 = 405
225 + 30w + w2 = 405

Factor a perfect square on the left side:
(w + 15)(w + 15) = 405

Calculate the square root of the right side: 20.124611797

Break this problem into two subproblems by setting 
(w + 15) equal to 20.124611797 and -20.124611797.

Subproblem 1

w + 15 = 20.124611797 Simplifying w + 15 = 20.124611797 Reorder the terms: 15 + w = 20.124611797 Solving 15 + w = 20.124611797 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + w = 20.124611797 + -15 Combine like terms: 15 + -15 = 0 0 + w = 20.124611797 + -15 w = 20.124611797 + -15 Combine like terms: 20.124611797 + -15 = 5.124611797 w = 5.124611797 Simplifying w = 5.124611797

Subproblem 2

w + 15 = -20.124611797 Simplifying w + 15 = -20.124611797 Reorder the terms: 15 + w = -20.124611797 Solving 15 + w = -20.124611797 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + w = -20.124611797 + -15 Combine like terms: 15 + -15 = 0 0 + w = -20.124611797 + -15 w = -20.124611797 + -15 Combine like terms: -20.124611797 + -15 = -35.124611797 w = -35.124611797 Simplifying w = -35.124611797

Solution

The solution to the problem is based on the solutions from the subproblems. w = {5.124611797, -35.124611797}

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