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Simplifying (4w + 10) * w = 103 Reorder the terms: (10 + 4w) * w = 103 Reorder the terms for easier multiplication: w(10 + 4w) = 103 (10 * w + 4w * w) = 103 (10w + 4w2) = 103 Solving 10w + 4w2 = 103 Solving for variable 'w'. Reorder the terms: -103 + 10w + 4w2 = 103 + -103 Combine like terms: 103 + -103 = 0 -103 + 10w + 4w2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -25.75 + 2.5w + w2 = 0 Move the constant term to the right: Add '25.75' to each side of the equation. -25.75 + 2.5w + 25.75 + w2 = 0 + 25.75 Reorder the terms: -25.75 + 25.75 + 2.5w + w2 = 0 + 25.75 Combine like terms: -25.75 + 25.75 = 0.00 0.00 + 2.5w + w2 = 0 + 25.75 2.5w + w2 = 0 + 25.75 Combine like terms: 0 + 25.75 = 25.75 2.5w + w2 = 25.75 The w term is 2.5w. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5w + 1.5625 + w2 = 25.75 + 1.5625 Reorder the terms: 1.5625 + 2.5w + w2 = 25.75 + 1.5625 Combine like terms: 25.75 + 1.5625 = 27.3125 1.5625 + 2.5w + w2 = 27.3125 Factor a perfect square on the left side: (w + 1.25)(w + 1.25) = 27.3125 Calculate the square root of the right side: 5.22613624 Break this problem into two subproblems by setting (w + 1.25) equal to 5.22613624 and -5.22613624.Subproblem 1
w + 1.25 = 5.22613624 Simplifying w + 1.25 = 5.22613624 Reorder the terms: 1.25 + w = 5.22613624 Solving 1.25 + w = 5.22613624 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + w = 5.22613624 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + w = 5.22613624 + -1.25 w = 5.22613624 + -1.25 Combine like terms: 5.22613624 + -1.25 = 3.97613624 w = 3.97613624 Simplifying w = 3.97613624Subproblem 2
w + 1.25 = -5.22613624 Simplifying w + 1.25 = -5.22613624 Reorder the terms: 1.25 + w = -5.22613624 Solving 1.25 + w = -5.22613624 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + w = -5.22613624 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + w = -5.22613624 + -1.25 w = -5.22613624 + -1.25 Combine like terms: -5.22613624 + -1.25 = -6.47613624 w = -6.47613624 Simplifying w = -6.47613624Solution
The solution to the problem is based on the solutions from the subproblems. w = {3.97613624, -6.47613624}
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