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Simplifying 2w2 + w + -90 = 0 Reorder the terms: -90 + w + 2w2 = 0 Solving -90 + w + 2w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -45 + 0.5w + w2 = 0 Move the constant term to the right: Add '45' to each side of the equation. -45 + 0.5w + 45 + w2 = 0 + 45 Reorder the terms: -45 + 45 + 0.5w + w2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + 0.5w + w2 = 0 + 45 0.5w + w2 = 0 + 45 Combine like terms: 0 + 45 = 45 0.5w + w2 = 45 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 = 45 + 0.25 Reorder the terms: 0.25 + 0.5w + w2 = 45 + 0.25 Combine like terms: 45 + 0.25 = 45.25 0.25 + 0.5w + w2 = 45.25 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 45.25 Calculate the square root of the right side: 6.726812024 Break this problem into two subproblems by setting (w + 0.5) equal to 6.726812024 and -6.726812024.Subproblem 1
w + 0.5 = 6.726812024 Simplifying w + 0.5 = 6.726812024 Reorder the terms: 0.5 + w = 6.726812024 Solving 0.5 + w = 6.726812024 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 = 6.726812024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 6.726812024 + -0.5 w = 6.726812024 + -0.5 Combine like terms: 6.726812024 + -0.5 = 6.226812024 w = 6.226812024 Simplifying w = 6.226812024Subproblem 2
w + 0.5 = -6.726812024 Simplifying w + 0.5 = -6.726812024 Reorder the terms: 0.5 + w = -6.726812024 Solving 0.5 + w = -6.726812024 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 = -6.726812024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -6.726812024 + -0.5 w = -6.726812024 + -0.5 Combine like terms: -6.726812024 + -0.5 = -7.226812024 w = -7.226812024 Simplifying w = -7.226812024Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.226812024, -7.226812024}
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