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Simplifying w(2w + 3) = 3 Reorder the terms: w(3 + 2w) = 3 (3 * w + 2w * w) = 3 (3w + 2w2) = 3 Solving 3w + 2w2 = 3 Solving for variable 'w'. Reorder the terms: -3 + 3w + 2w2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 3w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + 1.5w + w2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 1.5w + 1.5 + w2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 1.5w + w2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 1.5w + w2 = 0 + 1.5 1.5w + w2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 1.5w + w2 = 1.5 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 = 1.5 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = 1.5 + 0.5625 Combine like terms: 1.5 + 0.5625 = 2.0625 0.5625 + 1.5w + w2 = 2.0625 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (w + 0.75) equal to 1.436140662 and -1.436140662.Subproblem 1
w + 0.75 = 1.436140662 Simplifying w + 0.75 = 1.436140662 Reorder the terms: 0.75 + w = 1.436140662 Solving 0.75 + w = 1.436140662 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 = 1.436140662 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 1.436140662 + -0.75 w = 1.436140662 + -0.75 Combine like terms: 1.436140662 + -0.75 = 0.686140662 w = 0.686140662 Simplifying w = 0.686140662Subproblem 2
w + 0.75 = -1.436140662 Simplifying w + 0.75 = -1.436140662 Reorder the terms: 0.75 + w = -1.436140662 Solving 0.75 + w = -1.436140662 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 = -1.436140662 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -1.436140662 + -0.75 w = -1.436140662 + -0.75 Combine like terms: -1.436140662 + -0.75 = -2.186140662 w = -2.186140662 Simplifying w = -2.186140662Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.686140662, -2.186140662}
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