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Simplifying 2w2 + 11w + 23 + 2w + 10w + 25 = 0 Reorder the terms: 23 + 25 + 11w + 2w + 10w + 2w2 = 0 Combine like terms: 23 + 25 = 48 48 + 11w + 2w + 10w + 2w2 = 0 Combine like terms: 11w + 2w = 13w 48 + 13w + 10w + 2w2 = 0 Combine like terms: 13w + 10w = 23w 48 + 23w + 2w2 = 0 Solving 48 + 23w + 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'. 24 + 11.5w + w2 = 0 Move the constant term to the right: Add '-24' to each side of the equation. 24 + 11.5w + -24 + w2 = 0 + -24 Reorder the terms: 24 + -24 + 11.5w + w2 = 0 + -24 Combine like terms: 24 + -24 = 0 0 + 11.5w + w2 = 0 + -24 11.5w + w2 = 0 + -24 Combine like terms: 0 + -24 = -24 11.5w + w2 = -24 The w term is 11.5w. Take half its coefficient (5.75). Square it (33.0625) and add it to both sides. Add '33.0625' to each side of the equation. 11.5w + 33.0625 + w2 = -24 + 33.0625 Reorder the terms: 33.0625 + 11.5w + w2 = -24 + 33.0625 Combine like terms: -24 + 33.0625 = 9.0625 33.0625 + 11.5w + w2 = 9.0625 Factor a perfect square on the left side: (w + 5.75)(w + 5.75) = 9.0625 Calculate the square root of the right side: 3.010398645 Break this problem into two subproblems by setting (w + 5.75) equal to 3.010398645 and -3.010398645.Subproblem 1
w + 5.75 = 3.010398645 Simplifying w + 5.75 = 3.010398645 Reorder the terms: 5.75 + w = 3.010398645 Solving 5.75 + w = 3.010398645 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + w = 3.010398645 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + w = 3.010398645 + -5.75 w = 3.010398645 + -5.75 Combine like terms: 3.010398645 + -5.75 = -2.739601355 w = -2.739601355 Simplifying w = -2.739601355Subproblem 2
w + 5.75 = -3.010398645 Simplifying w + 5.75 = -3.010398645 Reorder the terms: 5.75 + w = -3.010398645 Solving 5.75 + w = -3.010398645 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + w = -3.010398645 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + w = -3.010398645 + -5.75 w = -3.010398645 + -5.75 Combine like terms: -3.010398645 + -5.75 = -8.760398645 w = -8.760398645 Simplifying w = -8.760398645Solution
The solution to the problem is based on the solutions from the subproblems. w = {-2.739601355, -8.760398645}
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