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Simplifying 9w2 + 30w = 25 Reorder the terms: 30w + 9w2 = 25 Solving 30w + 9w2 = 25 Solving for variable 'w'. Reorder the terms: -25 + 30w + 9w2 = 25 + -25 Combine like terms: 25 + -25 = 0 -25 + 30w + 9w2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.777777778 + 3.333333333w + w2 = 0 Move the constant term to the right: Add '2.777777778' to each side of the equation. -2.777777778 + 3.333333333w + 2.777777778 + w2 = 0 + 2.777777778 Reorder the terms: -2.777777778 + 2.777777778 + 3.333333333w + w2 = 0 + 2.777777778 Combine like terms: -2.777777778 + 2.777777778 = 0.000000000 0.000000000 + 3.333333333w + w2 = 0 + 2.777777778 3.333333333w + w2 = 0 + 2.777777778 Combine like terms: 0 + 2.777777778 = 2.777777778 3.333333333w + w2 = 2.777777778 The w term is 3.333333333w. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333w + 2.777777779 + w2 = 2.777777778 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333w + w2 = 2.777777778 + 2.777777779 Combine like terms: 2.777777778 + 2.777777779 = 5.555555557 2.777777779 + 3.333333333w + w2 = 5.555555557 Factor a perfect square on the left side: (w + 1.666666667)(w + 1.666666667) = 5.555555557 Calculate the square root of the right side: 2.357022604 Break this problem into two subproblems by setting (w + 1.666666667) equal to 2.357022604 and -2.357022604.Subproblem 1
w + 1.666666667 = 2.357022604 Simplifying w + 1.666666667 = 2.357022604 Reorder the terms: 1.666666667 + w = 2.357022604 Solving 1.666666667 + w = 2.357022604 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + w = 2.357022604 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + w = 2.357022604 + -1.666666667 w = 2.357022604 + -1.666666667 Combine like terms: 2.357022604 + -1.666666667 = 0.690355937 w = 0.690355937 Simplifying w = 0.690355937Subproblem 2
w + 1.666666667 = -2.357022604 Simplifying w + 1.666666667 = -2.357022604 Reorder the terms: 1.666666667 + w = -2.357022604 Solving 1.666666667 + w = -2.357022604 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + w = -2.357022604 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + w = -2.357022604 + -1.666666667 w = -2.357022604 + -1.666666667 Combine like terms: -2.357022604 + -1.666666667 = -4.023689271 w = -4.023689271 Simplifying w = -4.023689271Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.690355937, -4.023689271}
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