(2w+1)(w)=88

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Solution for (2w+1)(w)=88 equation:


Simplifying
(2w + 1)(w) = 88

Reorder the terms:
(1 + 2w)(w) = 88

Reorder the terms for easier multiplication:
w(1 + 2w) = 88
(1 * w + 2w * w) = 88
(1w + 2w2) = 88

Solving
1w + 2w2 = 88

Solving for variable 'w'.

Reorder the terms:
-88 + 1w + 2w2 = 88 + -88

Combine like terms: 88 + -88 = 0
-88 + 1w + 2w2 = 0

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

Divide each side by '2'.
-44 + 0.5w + w2 = 0

Move the constant term to the right:

Add '44' to each side of the equation.
-44 + 0.5w + 44 + w2 = 0 + 44

Reorder the terms:
-44 + 44 + 0.5w + w2 = 0 + 44

Combine like terms: -44 + 44 = 0
0 + 0.5w + w2 = 0 + 44
0.5w + w2 = 0 + 44

Combine like terms: 0 + 44 = 44
0.5w + w2 = 44

The w term is 0.5w.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5w + 0.0625 + w2 = 44 + 0.0625

Reorder the terms:
0.0625 + 0.5w + w2 = 44 + 0.0625

Combine like terms: 44 + 0.0625 = 44.0625
0.0625 + 0.5w + w2 = 44.0625

Factor a perfect square on the left side:
(w + 0.25)(w + 0.25) = 44.0625

Calculate the square root of the right side: 6.637959024

Break this problem into two subproblems by setting 
(w + 0.25) equal to 6.637959024 and -6.637959024.

Subproblem 1

w + 0.25 = 6.637959024 Simplifying w + 0.25 = 6.637959024 Reorder the terms: 0.25 + w = 6.637959024 Solving 0.25 + w = 6.637959024 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + w = 6.637959024 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = 6.637959024 + -0.25 w = 6.637959024 + -0.25 Combine like terms: 6.637959024 + -0.25 = 6.387959024 w = 6.387959024 Simplifying w = 6.387959024

Subproblem 2

w + 0.25 = -6.637959024 Simplifying w + 0.25 = -6.637959024 Reorder the terms: 0.25 + w = -6.637959024 Solving 0.25 + w = -6.637959024 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + w = -6.637959024 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = -6.637959024 + -0.25 w = -6.637959024 + -0.25 Combine like terms: -6.637959024 + -0.25 = -6.887959024 w = -6.887959024 Simplifying w = -6.887959024

Solution

The solution to the problem is based on the solutions from the subproblems. w = {6.387959024, -6.887959024}

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