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Simplifying (3.5 + w) * w = 31 Reorder the terms for easier multiplication: w(3.5 + w) = 31 (3.5 * w + w * w) = 31 (3.5w + w2) = 31 Solving 3.5w + w2 = 31 Solving for variable 'w'. Reorder the terms: -31 + 3.5w + w2 = 31 + -31 Combine like terms: 31 + -31 = 0 -31 + 3.5w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '31' to each side of the equation. -31 + 3.5w + 31 + w2 = 0 + 31 Reorder the terms: -31 + 31 + 3.5w + w2 = 0 + 31 Combine like terms: -31 + 31 = 0 0 + 3.5w + w2 = 0 + 31 3.5w + w2 = 0 + 31 Combine like terms: 0 + 31 = 31 3.5w + w2 = 31 The w term is 3.5w. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5w + 3.0625 + w2 = 31 + 3.0625 Reorder the terms: 3.0625 + 3.5w + w2 = 31 + 3.0625 Combine like terms: 31 + 3.0625 = 34.0625 3.0625 + 3.5w + w2 = 34.0625 Factor a perfect square on the left side: (w + 1.75)(w + 1.75) = 34.0625 Calculate the square root of the right side: 5.836308765 Break this problem into two subproblems by setting (w + 1.75) equal to 5.836308765 and -5.836308765.Subproblem 1
w + 1.75 = 5.836308765 Simplifying w + 1.75 = 5.836308765 Reorder the terms: 1.75 + w = 5.836308765 Solving 1.75 + w = 5.836308765 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + w = 5.836308765 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + w = 5.836308765 + -1.75 w = 5.836308765 + -1.75 Combine like terms: 5.836308765 + -1.75 = 4.086308765 w = 4.086308765 Simplifying w = 4.086308765Subproblem 2
w + 1.75 = -5.836308765 Simplifying w + 1.75 = -5.836308765 Reorder the terms: 1.75 + w = -5.836308765 Solving 1.75 + w = -5.836308765 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + w = -5.836308765 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + w = -5.836308765 + -1.75 w = -5.836308765 + -1.75 Combine like terms: -5.836308765 + -1.75 = -7.586308765 w = -7.586308765 Simplifying w = -7.586308765Solution
The solution to the problem is based on the solutions from the subproblems. w = {4.086308765, -7.586308765}
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