241=2(w+16)w

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Solution for 241=2(w+16)w equation:


Simplifying
241 = 2(w + 16) * w

Reorder the terms:
241 = 2(16 + w) * w

Reorder the terms for easier multiplication:
241 = 2w(16 + w)
241 = (16 * 2w + w * 2w)
241 = (32w + 2w2)

Solving
241 = 32w + 2w2

Solving for variable 'w'.

Reorder the terms:
241 + -32w + -2w2 = 32w + -32w + 2w2 + -2w2

Combine like terms: 32w + -32w = 0
241 + -32w + -2w2 = 0 + 2w2 + -2w2
241 + -32w + -2w2 = 2w2 + -2w2

Combine like terms: 2w2 + -2w2 = 0
241 + -32w + -2w2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-120.5 + 16w + w2 = 0

Move the constant term to the right:

Add '120.5' to each side of the equation.
-120.5 + 16w + 120.5 + w2 = 0 + 120.5

Reorder the terms:
-120.5 + 120.5 + 16w + w2 = 0 + 120.5

Combine like terms: -120.5 + 120.5 = 0.0
0.0 + 16w + w2 = 0 + 120.5
16w + w2 = 0 + 120.5

Combine like terms: 0 + 120.5 = 120.5
16w + w2 = 120.5

The w term is 16w.  Take half its coefficient (8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
16w + 64 + w2 = 120.5 + 64

Reorder the terms:
64 + 16w + w2 = 120.5 + 64

Combine like terms: 120.5 + 64 = 184.5
64 + 16w + w2 = 184.5

Factor a perfect square on the left side:
(w + 8)(w + 8) = 184.5

Calculate the square root of the right side: 13.583077707

Break this problem into two subproblems by setting 
(w + 8) equal to 13.583077707 and -13.583077707.

Subproblem 1

w + 8 = 13.583077707 Simplifying w + 8 = 13.583077707 Reorder the terms: 8 + w = 13.583077707 Solving 8 + w = 13.583077707 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + w = 13.583077707 + -8 Combine like terms: 8 + -8 = 0 0 + w = 13.583077707 + -8 w = 13.583077707 + -8 Combine like terms: 13.583077707 + -8 = 5.583077707 w = 5.583077707 Simplifying w = 5.583077707

Subproblem 2

w + 8 = -13.583077707 Simplifying w + 8 = -13.583077707 Reorder the terms: 8 + w = -13.583077707 Solving 8 + w = -13.583077707 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + w = -13.583077707 + -8 Combine like terms: 8 + -8 = 0 0 + w = -13.583077707 + -8 w = -13.583077707 + -8 Combine like terms: -13.583077707 + -8 = -21.583077707 w = -21.583077707 Simplifying w = -21.583077707

Solution

The solution to the problem is based on the solutions from the subproblems. w = {5.583077707, -21.583077707}

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