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Simplifying (9w + 8)(1w + -10) = w Reorder the terms: (8 + 9w)(1w + -10) = w Reorder the terms: (8 + 9w)(-10 + 1w) = w Multiply (8 + 9w) * (-10 + 1w) (8(-10 + 1w) + 9w * (-10 + 1w)) = w ((-10 * 8 + 1w * 8) + 9w * (-10 + 1w)) = w ((-80 + 8w) + 9w * (-10 + 1w)) = w (-80 + 8w + (-10 * 9w + 1w * 9w)) = w (-80 + 8w + (-90w + 9w2)) = w Combine like terms: 8w + -90w = -82w (-80 + -82w + 9w2) = w Solving -80 + -82w + 9w2 = w Solving for variable 'w'. Reorder the terms: -80 + -82w + -1w + 9w2 = w + -1w Combine like terms: -82w + -1w = -83w -80 + -83w + 9w2 = w + -1w Combine like terms: w + -1w = 0 -80 + -83w + 9w2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -8.888888889 + -9.222222222w + w2 = 0 Move the constant term to the right: Add '8.888888889' to each side of the equation. -8.888888889 + -9.222222222w + 8.888888889 + w2 = 0 + 8.888888889 Reorder the terms: -8.888888889 + 8.888888889 + -9.222222222w + w2 = 0 + 8.888888889 Combine like terms: -8.888888889 + 8.888888889 = 0.000000000 0.000000000 + -9.222222222w + w2 = 0 + 8.888888889 -9.222222222w + w2 = 0 + 8.888888889 Combine like terms: 0 + 8.888888889 = 8.888888889 -9.222222222w + w2 = 8.888888889 The w term is -9.222222222w. Take half its coefficient (-4.611111111). Square it (21.26234568) and add it to both sides. Add '21.26234568' to each side of the equation. -9.222222222w + 21.26234568 + w2 = 8.888888889 + 21.26234568 Reorder the terms: 21.26234568 + -9.222222222w + w2 = 8.888888889 + 21.26234568 Combine like terms: 8.888888889 + 21.26234568 = 30.151234569 21.26234568 + -9.222222222w + w2 = 30.151234569 Factor a perfect square on the left side: (w + -4.611111111)(w + -4.611111111) = 30.151234569 Calculate the square root of the right side: 5.491013984 Break this problem into two subproblems by setting (w + -4.611111111) equal to 5.491013984 and -5.491013984.Subproblem 1
w + -4.611111111 = 5.491013984 Simplifying w + -4.611111111 = 5.491013984 Reorder the terms: -4.611111111 + w = 5.491013984 Solving -4.611111111 + w = 5.491013984 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '4.611111111' to each side of the equation. -4.611111111 + 4.611111111 + w = 5.491013984 + 4.611111111 Combine like terms: -4.611111111 + 4.611111111 = 0.000000000 0.000000000 + w = 5.491013984 + 4.611111111 w = 5.491013984 + 4.611111111 Combine like terms: 5.491013984 + 4.611111111 = 10.102125095 w = 10.102125095 Simplifying w = 10.102125095Subproblem 2
w + -4.611111111 = -5.491013984 Simplifying w + -4.611111111 = -5.491013984 Reorder the terms: -4.611111111 + w = -5.491013984 Solving -4.611111111 + w = -5.491013984 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '4.611111111' to each side of the equation. -4.611111111 + 4.611111111 + w = -5.491013984 + 4.611111111 Combine like terms: -4.611111111 + 4.611111111 = 0.000000000 0.000000000 + w = -5.491013984 + 4.611111111 w = -5.491013984 + 4.611111111 Combine like terms: -5.491013984 + 4.611111111 = -0.879902873 w = -0.879902873 Simplifying w = -0.879902873Solution
The solution to the problem is based on the solutions from the subproblems. w = {10.102125095, -0.879902873}
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