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Simplifying 2w2 + 4 + w + 6 = 480 Reorder the terms: 4 + 6 + w + 2w2 = 480 Combine like terms: 4 + 6 = 10 10 + w + 2w2 = 480 Solving 10 + w + 2w2 = 480 Solving for variable 'w'. Reorder the terms: 10 + -480 + w + 2w2 = 480 + -480 Combine like terms: 10 + -480 = -470 -470 + w + 2w2 = 480 + -480 Combine like terms: 480 + -480 = 0 -470 + w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -235 + 0.5w + w2 = 0 Move the constant term to the right: Add '235' to each side of the equation. -235 + 0.5w + 235 + w2 = 0 + 235 Reorder the terms: -235 + 235 + 0.5w + w2 = 0 + 235 Combine like terms: -235 + 235 = 0 0 + 0.5w + w2 = 0 + 235 0.5w + w2 = 0 + 235 Combine like terms: 0 + 235 = 235 0.5w + w2 = 235 The w term is w. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5w + 0.25 + w2 = 235 + 0.25 Reorder the terms: 0.25 + 0.5w + w2 = 235 + 0.25 Combine like terms: 235 + 0.25 = 235.25 0.25 + 0.5w + w2 = 235.25 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 235.25 Calculate the square root of the right side: 15.33786165 Break this problem into two subproblems by setting (w + 0.5) equal to 15.33786165 and -15.33786165.Subproblem 1
w + 0.5 = 15.33786165 Simplifying w + 0.5 = 15.33786165 Reorder the terms: 0.5 + w = 15.33786165 Solving 0.5 + w = 15.33786165 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = 15.33786165 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 15.33786165 + -0.5 w = 15.33786165 + -0.5 Combine like terms: 15.33786165 + -0.5 = 14.83786165 w = 14.83786165 Simplifying w = 14.83786165Subproblem 2
w + 0.5 = -15.33786165 Simplifying w + 0.5 = -15.33786165 Reorder the terms: 0.5 + w = -15.33786165 Solving 0.5 + w = -15.33786165 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = -15.33786165 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -15.33786165 + -0.5 w = -15.33786165 + -0.5 Combine like terms: -15.33786165 + -0.5 = -15.83786165 w = -15.83786165 Simplifying w = -15.83786165Solution
The solution to the problem is based on the solutions from the subproblems. w = {14.83786165, -15.83786165}
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