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Simplifying 0 = 2w2 + w + -5 Reorder the terms: 0 = -5 + w + 2w2 Solving 0 = -5 + w + 2w2 Solving for variable 'w'. Combine like terms: 0 + 5 = 5 5 + -1w + -2w2 = -5 + w + 2w2 + 5 + -1w + -2w2 Reorder the terms: 5 + -1w + -2w2 = -5 + 5 + w + -1w + 2w2 + -2w2 Combine like terms: -5 + 5 = 0 5 + -1w + -2w2 = 0 + w + -1w + 2w2 + -2w2 5 + -1w + -2w2 = w + -1w + 2w2 + -2w2 Combine like terms: w + -1w = 0 5 + -1w + -2w2 = 0 + 2w2 + -2w2 5 + -1w + -2w2 = 2w2 + -2w2 Combine like terms: 2w2 + -2w2 = 0 5 + -1w + -2w2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.5 + 0.5w + w2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 0.5w + 2.5 + w2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 0.5w + w2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 0.5w + w2 = 0 + 2.5 0.5w + w2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 0.5w + w2 = 2.5 The w term is 0.5w. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5w + 0.0625 + w2 = 2.5 + 0.0625 Reorder the terms: 0.0625 + 0.5w + w2 = 2.5 + 0.0625 Combine like terms: 2.5 + 0.0625 = 2.5625 0.0625 + 0.5w + w2 = 2.5625 Factor a perfect square on the left side: (w + 0.25)(w + 0.25) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (w + 0.25) equal to 1.600781059 and -1.600781059.Subproblem 1
w + 0.25 = 1.600781059 Simplifying w + 0.25 = 1.600781059 Reorder the terms: 0.25 + w = 1.600781059 Solving 0.25 + w = 1.600781059 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + w = 1.600781059 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = 1.600781059 + -0.25 w = 1.600781059 + -0.25 Combine like terms: 1.600781059 + -0.25 = 1.350781059 w = 1.350781059 Simplifying w = 1.350781059Subproblem 2
w + 0.25 = -1.600781059 Simplifying w + 0.25 = -1.600781059 Reorder the terms: 0.25 + w = -1.600781059 Solving 0.25 + w = -1.600781059 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + w = -1.600781059 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = -1.600781059 + -0.25 w = -1.600781059 + -0.25 Combine like terms: -1.600781059 + -0.25 = -1.850781059 w = -1.850781059 Simplifying w = -1.850781059Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.350781059, -1.850781059}
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