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Simplifying 36 = (3 + 2w) * w Reorder the terms for easier multiplication: 36 = w(3 + 2w) 36 = (3 * w + 2w * w) 36 = (3w + 2w2) Solving 36 = 3w + 2w2 Solving for variable 'w'. Reorder the terms: 36 + -3w + -2w2 = 3w + -3w + 2w2 + -2w2 Combine like terms: 3w + -3w = 0 36 + -3w + -2w2 = 0 + 2w2 + -2w2 36 + -3w + -2w2 = 2w2 + -2w2 Combine like terms: 2w2 + -2w2 = 0 36 + -3w + -2w2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -18 + 1.5w + w2 = 0 Move the constant term to the right: Add '18' to each side of the equation. -18 + 1.5w + 18 + w2 = 0 + 18 Reorder the terms: -18 + 18 + 1.5w + w2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 1.5w + w2 = 0 + 18 1.5w + w2 = 0 + 18 Combine like terms: 0 + 18 = 18 1.5w + w2 = 18 The w term is 1.5w. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5w + 0.5625 + w2 = 18 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = 18 + 0.5625 Combine like terms: 18 + 0.5625 = 18.5625 0.5625 + 1.5w + w2 = 18.5625 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 18.5625 Calculate the square root of the right side: 4.308421985 Break this problem into two subproblems by setting (w + 0.75) equal to 4.308421985 and -4.308421985.Subproblem 1
w + 0.75 = 4.308421985 Simplifying w + 0.75 = 4.308421985 Reorder the terms: 0.75 + w = 4.308421985 Solving 0.75 + w = 4.308421985 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = 4.308421985 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 4.308421985 + -0.75 w = 4.308421985 + -0.75 Combine like terms: 4.308421985 + -0.75 = 3.558421985 w = 3.558421985 Simplifying w = 3.558421985Subproblem 2
w + 0.75 = -4.308421985 Simplifying w + 0.75 = -4.308421985 Reorder the terms: 0.75 + w = -4.308421985 Solving 0.75 + w = -4.308421985 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = -4.308421985 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -4.308421985 + -0.75 w = -4.308421985 + -0.75 Combine like terms: -4.308421985 + -0.75 = -5.058421985 w = -5.058421985 Simplifying w = -5.058421985Solution
The solution to the problem is based on the solutions from the subproblems. w = {3.558421985, -5.058421985}
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