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Simplifying 5w2 + 2w2 + 25w + 10 = 0 Reorder the terms: 10 + 25w + 5w2 + 2w2 = 0 Combine like terms: 5w2 + 2w2 = 7w2 10 + 25w + 7w2 = 0 Solving 10 + 25w + 7w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 1.428571429 + 3.571428571w + w2 = 0 Move the constant term to the right: Add '-1.428571429' to each side of the equation. 1.428571429 + 3.571428571w + -1.428571429 + w2 = 0 + -1.428571429 Reorder the terms: 1.428571429 + -1.428571429 + 3.571428571w + w2 = 0 + -1.428571429 Combine like terms: 1.428571429 + -1.428571429 = 0.000000000 0.000000000 + 3.571428571w + w2 = 0 + -1.428571429 3.571428571w + w2 = 0 + -1.428571429 Combine like terms: 0 + -1.428571429 = -1.428571429 3.571428571w + w2 = -1.428571429 The w term is 3.571428571w. Take half its coefficient (1.785714286). Square it (3.188775511) and add it to both sides. Add '3.188775511' to each side of the equation. 3.571428571w + 3.188775511 + w2 = -1.428571429 + 3.188775511 Reorder the terms: 3.188775511 + 3.571428571w + w2 = -1.428571429 + 3.188775511 Combine like terms: -1.428571429 + 3.188775511 = 1.760204082 3.188775511 + 3.571428571w + w2 = 1.760204082 Factor a perfect square on the left side: (w + 1.785714286)(w + 1.785714286) = 1.760204082 Calculate the square root of the right side: 1.32672683 Break this problem into two subproblems by setting (w + 1.785714286) equal to 1.32672683 and -1.32672683.Subproblem 1
w + 1.785714286 = 1.32672683 Simplifying w + 1.785714286 = 1.32672683 Reorder the terms: 1.785714286 + w = 1.32672683 Solving 1.785714286 + w = 1.32672683 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.785714286' to each side of the equation. 1.785714286 + -1.785714286 + w = 1.32672683 + -1.785714286 Combine like terms: 1.785714286 + -1.785714286 = 0.000000000 0.000000000 + w = 1.32672683 + -1.785714286 w = 1.32672683 + -1.785714286 Combine like terms: 1.32672683 + -1.785714286 = -0.458987456 w = -0.458987456 Simplifying w = -0.458987456Subproblem 2
w + 1.785714286 = -1.32672683 Simplifying w + 1.785714286 = -1.32672683 Reorder the terms: 1.785714286 + w = -1.32672683 Solving 1.785714286 + w = -1.32672683 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.785714286' to each side of the equation. 1.785714286 + -1.785714286 + w = -1.32672683 + -1.785714286 Combine like terms: 1.785714286 + -1.785714286 = 0.000000000 0.000000000 + w = -1.32672683 + -1.785714286 w = -1.32672683 + -1.785714286 Combine like terms: -1.32672683 + -1.785714286 = -3.112441116 w = -3.112441116 Simplifying w = -3.112441116Solution
The solution to the problem is based on the solutions from the subproblems. w = {-0.458987456, -3.112441116}
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