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Simplifying 22 = (2w + 3.5) * w Reorder the terms: 22 = (3.5 + 2w) * w Reorder the terms for easier multiplication: 22 = w(3.5 + 2w) 22 = (3.5 * w + 2w * w) 22 = (3.5w + 2w2) Solving 22 = 3.5w + 2w2 Solving for variable 'w'. Reorder the terms: 22 + -3.5w + -2w2 = 3.5w + -3.5w + 2w2 + -2w2 Combine like terms: 3.5w + -3.5w = 0.0 22 + -3.5w + -2w2 = 0.0 + 2w2 + -2w2 22 + -3.5w + -2w2 = 2w2 + -2w2 Combine like terms: 2w2 + -2w2 = 0 22 + -3.5w + -2w2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -11 + 1.75w + w2 = 0 Move the constant term to the right: Add '11' to each side of the equation. -11 + 1.75w + 11 + w2 = 0 + 11 Reorder the terms: -11 + 11 + 1.75w + w2 = 0 + 11 Combine like terms: -11 + 11 = 0 0 + 1.75w + w2 = 0 + 11 1.75w + w2 = 0 + 11 Combine like terms: 0 + 11 = 11 1.75w + w2 = 11 The w term is 1.75w. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75w + 0.765625 + w2 = 11 + 0.765625 Reorder the terms: 0.765625 + 1.75w + w2 = 11 + 0.765625 Combine like terms: 11 + 0.765625 = 11.765625 0.765625 + 1.75w + w2 = 11.765625 Factor a perfect square on the left side: (w + 0.875)(w + 0.875) = 11.765625 Calculate the square root of the right side: 3.430105684 Break this problem into two subproblems by setting (w + 0.875) equal to 3.430105684 and -3.430105684.Subproblem 1
w + 0.875 = 3.430105684 Simplifying w + 0.875 = 3.430105684 Reorder the terms: 0.875 + w = 3.430105684 Solving 0.875 + w = 3.430105684 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + w = 3.430105684 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + w = 3.430105684 + -0.875 w = 3.430105684 + -0.875 Combine like terms: 3.430105684 + -0.875 = 2.555105684 w = 2.555105684 Simplifying w = 2.555105684Subproblem 2
w + 0.875 = -3.430105684 Simplifying w + 0.875 = -3.430105684 Reorder the terms: 0.875 + w = -3.430105684 Solving 0.875 + w = -3.430105684 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + w = -3.430105684 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + w = -3.430105684 + -0.875 w = -3.430105684 + -0.875 Combine like terms: -3.430105684 + -0.875 = -4.305105684 w = -4.305105684 Simplifying w = -4.305105684Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.555105684, -4.305105684}
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