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Simplifying -1w2 + 2w + 5w2 + -3 = 0 Reorder the terms: -3 + 2w + -1w2 + 5w2 = 0 Combine like terms: -1w2 + 5w2 = 4w2 -3 + 2w + 4w2 = 0 Solving -3 + 2w + 4w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + 0.5w + w2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 0.5w + 0.75 + w2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 0.5w + w2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 0.5w + w2 = 0 + 0.75 0.5w + w2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 0.5w + w2 = 0.75 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 = 0.75 + 0.0625 Reorder the terms: 0.0625 + 0.5w + w2 = 0.75 + 0.0625 Combine like terms: 0.75 + 0.0625 = 0.8125 0.0625 + 0.5w + w2 = 0.8125 Factor a perfect square on the left side: (w + 0.25)(w + 0.25) = 0.8125 Calculate the square root of the right side: 0.901387819 Break this problem into two subproblems by setting (w + 0.25) equal to 0.901387819 and -0.901387819.Subproblem 1
w + 0.25 = 0.901387819 Simplifying w + 0.25 = 0.901387819 Reorder the terms: 0.25 + w = 0.901387819 Solving 0.25 + w = 0.901387819 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 = 0.901387819 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = 0.901387819 + -0.25 w = 0.901387819 + -0.25 Combine like terms: 0.901387819 + -0.25 = 0.651387819 w = 0.651387819 Simplifying w = 0.651387819Subproblem 2
w + 0.25 = -0.901387819 Simplifying w + 0.25 = -0.901387819 Reorder the terms: 0.25 + w = -0.901387819 Solving 0.25 + w = -0.901387819 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 = -0.901387819 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = -0.901387819 + -0.25 w = -0.901387819 + -0.25 Combine like terms: -0.901387819 + -0.25 = -1.151387819 w = -1.151387819 Simplifying w = -1.151387819Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.651387819, -1.151387819}
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