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Simplifying (2w + 3) * w = 54 * 54 Reorder the terms: (3 + 2w) * w = 54 * 54 Reorder the terms for easier multiplication: w(3 + 2w) = 54 * 54 (3 * w + 2w * w) = 54 * 54 (3w + 2w2) = 54 * 54 Multiply 54 * 54 3w + 2w2 = 2916 Solving 3w + 2w2 = 2916 Solving for variable 'w'. Reorder the terms: -2916 + 3w + 2w2 = 2916 + -2916 Combine like terms: 2916 + -2916 = 0 -2916 + 3w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1458 + 1.5w + w2 = 0 Move the constant term to the right: Add '1458' to each side of the equation. -1458 + 1.5w + 1458 + w2 = 0 + 1458 Reorder the terms: -1458 + 1458 + 1.5w + w2 = 0 + 1458 Combine like terms: -1458 + 1458 = 0 0 + 1.5w + w2 = 0 + 1458 1.5w + w2 = 0 + 1458 Combine like terms: 0 + 1458 = 1458 1.5w + w2 = 1458 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 = 1458 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = 1458 + 0.5625 Combine like terms: 1458 + 0.5625 = 1458.5625 0.5625 + 1.5w + w2 = 1458.5625 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 1458.5625 Calculate the square root of the right side: 38.191131169 Break this problem into two subproblems by setting (w + 0.75) equal to 38.191131169 and -38.191131169.Subproblem 1
w + 0.75 = 38.191131169 Simplifying w + 0.75 = 38.191131169 Reorder the terms: 0.75 + w = 38.191131169 Solving 0.75 + w = 38.191131169 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 = 38.191131169 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 38.191131169 + -0.75 w = 38.191131169 + -0.75 Combine like terms: 38.191131169 + -0.75 = 37.441131169 w = 37.441131169 Simplifying w = 37.441131169Subproblem 2
w + 0.75 = -38.191131169 Simplifying w + 0.75 = -38.191131169 Reorder the terms: 0.75 + w = -38.191131169 Solving 0.75 + w = -38.191131169 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 = -38.191131169 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -38.191131169 + -0.75 w = -38.191131169 + -0.75 Combine like terms: -38.191131169 + -0.75 = -38.941131169 w = -38.941131169 Simplifying w = -38.941131169Solution
The solution to the problem is based on the solutions from the subproblems. w = {37.441131169, -38.941131169}
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