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Simplifying (8w2 + -5w + 4) + -4(w2 + 6w + -1) = 0 Reorder the terms: (4 + -5w + 8w2) + -4(w2 + 6w + -1) = 0 Remove parenthesis around (4 + -5w + 8w2) 4 + -5w + 8w2 + -4(w2 + 6w + -1) = 0 Reorder the terms: 4 + -5w + 8w2 + -4(-1 + 6w + w2) = 0 4 + -5w + 8w2 + (-1 * -4 + 6w * -4 + w2 * -4) = 0 4 + -5w + 8w2 + (4 + -24w + -4w2) = 0 Reorder the terms: 4 + 4 + -5w + -24w + 8w2 + -4w2 = 0 Combine like terms: 4 + 4 = 8 8 + -5w + -24w + 8w2 + -4w2 = 0 Combine like terms: -5w + -24w = -29w 8 + -29w + 8w2 + -4w2 = 0 Combine like terms: 8w2 + -4w2 = 4w2 8 + -29w + 4w2 = 0 Solving 8 + -29w + 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'. 2 + -7.25w + w2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -7.25w + -2 + w2 = 0 + -2 Reorder the terms: 2 + -2 + -7.25w + w2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -7.25w + w2 = 0 + -2 -7.25w + w2 = 0 + -2 Combine like terms: 0 + -2 = -2 -7.25w + w2 = -2 The w term is -7.25w. Take half its coefficient (-3.625). Square it (13.140625) and add it to both sides. Add '13.140625' to each side of the equation. -7.25w + 13.140625 + w2 = -2 + 13.140625 Reorder the terms: 13.140625 + -7.25w + w2 = -2 + 13.140625 Combine like terms: -2 + 13.140625 = 11.140625 13.140625 + -7.25w + w2 = 11.140625 Factor a perfect square on the left side: (w + -3.625)(w + -3.625) = 11.140625 Calculate the square root of the right side: 3.337757481 Break this problem into two subproblems by setting (w + -3.625) equal to 3.337757481 and -3.337757481.Subproblem 1
w + -3.625 = 3.337757481 Simplifying w + -3.625 = 3.337757481 Reorder the terms: -3.625 + w = 3.337757481 Solving -3.625 + w = 3.337757481 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '3.625' to each side of the equation. -3.625 + 3.625 + w = 3.337757481 + 3.625 Combine like terms: -3.625 + 3.625 = 0.000 0.000 + w = 3.337757481 + 3.625 w = 3.337757481 + 3.625 Combine like terms: 3.337757481 + 3.625 = 6.962757481 w = 6.962757481 Simplifying w = 6.962757481Subproblem 2
w + -3.625 = -3.337757481 Simplifying w + -3.625 = -3.337757481 Reorder the terms: -3.625 + w = -3.337757481 Solving -3.625 + w = -3.337757481 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '3.625' to each side of the equation. -3.625 + 3.625 + w = -3.337757481 + 3.625 Combine like terms: -3.625 + 3.625 = 0.000 0.000 + w = -3.337757481 + 3.625 w = -3.337757481 + 3.625 Combine like terms: -3.337757481 + 3.625 = 0.287242519 w = 0.287242519 Simplifying w = 0.287242519Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.962757481, 0.287242519}
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