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Simplifying 2w2 + 3w = 2.33 Reorder the terms: 3w + 2w2 = 2.33 Solving 3w + 2w2 = 2.33 Solving for variable 'w'. Reorder the terms: -2.33 + 3w + 2w2 = 2.33 + -2.33 Combine like terms: 2.33 + -2.33 = 0.00 -2.33 + 3w + 2w2 = 0.00 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.165 + 1.5w + w2 = 0.00 Move the constant term to the right: Add '1.165' to each side of the equation. -1.165 + 1.5w + 1.165 + w2 = 0.00 + 1.165 Reorder the terms: -1.165 + 1.165 + 1.5w + w2 = 0.00 + 1.165 Combine like terms: -1.165 + 1.165 = 0.000 0.000 + 1.5w + w2 = 0.00 + 1.165 1.5w + w2 = 0.00 + 1.165 Combine like terms: 0.00 + 1.165 = 1.165 1.5w + w2 = 1.165 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 = 1.165 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = 1.165 + 0.5625 Combine like terms: 1.165 + 0.5625 = 1.7275 0.5625 + 1.5w + w2 = 1.7275 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 1.7275 Calculate the square root of the right side: 1.314343943 Break this problem into two subproblems by setting (w + 0.75) equal to 1.314343943 and -1.314343943.Subproblem 1
w + 0.75 = 1.314343943 Simplifying w + 0.75 = 1.314343943 Reorder the terms: 0.75 + w = 1.314343943 Solving 0.75 + w = 1.314343943 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 = 1.314343943 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 1.314343943 + -0.75 w = 1.314343943 + -0.75 Combine like terms: 1.314343943 + -0.75 = 0.564343943 w = 0.564343943 Simplifying w = 0.564343943Subproblem 2
w + 0.75 = -1.314343943 Simplifying w + 0.75 = -1.314343943 Reorder the terms: 0.75 + w = -1.314343943 Solving 0.75 + w = -1.314343943 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 = -1.314343943 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -1.314343943 + -0.75 w = -1.314343943 + -0.75 Combine like terms: -1.314343943 + -0.75 = -2.064343943 w = -2.064343943 Simplifying w = -2.064343943Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.564343943, -2.064343943}
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