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Simplifying 2w + -9 + 24w2 + 3w + -14 = 0 Reorder the terms: -9 + -14 + 2w + 3w + 24w2 = 0 Combine like terms: -9 + -14 = -23 -23 + 2w + 3w + 24w2 = 0 Combine like terms: 2w + 3w = 5w -23 + 5w + 24w2 = 0 Solving -23 + 5w + 24w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -0.9583333333 + 0.2083333333w + w2 = 0 Move the constant term to the right: Add '0.9583333333' to each side of the equation. -0.9583333333 + 0.2083333333w + 0.9583333333 + w2 = 0 + 0.9583333333 Reorder the terms: -0.9583333333 + 0.9583333333 + 0.2083333333w + w2 = 0 + 0.9583333333 Combine like terms: -0.9583333333 + 0.9583333333 = 0.0000000000 0.0000000000 + 0.2083333333w + w2 = 0 + 0.9583333333 0.2083333333w + w2 = 0 + 0.9583333333 Combine like terms: 0 + 0.9583333333 = 0.9583333333 0.2083333333w + w2 = 0.9583333333 The w term is 0.2083333333w. Take half its coefficient (0.1041666667). Square it (0.01085069445) and add it to both sides. Add '0.01085069445' to each side of the equation. 0.2083333333w + 0.01085069445 + w2 = 0.9583333333 + 0.01085069445 Reorder the terms: 0.01085069445 + 0.2083333333w + w2 = 0.9583333333 + 0.01085069445 Combine like terms: 0.9583333333 + 0.01085069445 = 0.96918402775 0.01085069445 + 0.2083333333w + w2 = 0.96918402775 Factor a perfect square on the left side: (w + 0.1041666667)(w + 0.1041666667) = 0.96918402775 Calculate the square root of the right side: 0.984471446 Break this problem into two subproblems by setting (w + 0.1041666667) equal to 0.984471446 and -0.984471446.Subproblem 1
w + 0.1041666667 = 0.984471446 Simplifying w + 0.1041666667 = 0.984471446 Reorder the terms: 0.1041666667 + w = 0.984471446 Solving 0.1041666667 + w = 0.984471446 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.1041666667' to each side of the equation. 0.1041666667 + -0.1041666667 + w = 0.984471446 + -0.1041666667 Combine like terms: 0.1041666667 + -0.1041666667 = 0.0000000000 0.0000000000 + w = 0.984471446 + -0.1041666667 w = 0.984471446 + -0.1041666667 Combine like terms: 0.984471446 + -0.1041666667 = 0.8803047793 w = 0.8803047793 Simplifying w = 0.8803047793Subproblem 2
w + 0.1041666667 = -0.984471446 Simplifying w + 0.1041666667 = -0.984471446 Reorder the terms: 0.1041666667 + w = -0.984471446 Solving 0.1041666667 + w = -0.984471446 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.1041666667' to each side of the equation. 0.1041666667 + -0.1041666667 + w = -0.984471446 + -0.1041666667 Combine like terms: 0.1041666667 + -0.1041666667 = 0.0000000000 0.0000000000 + w = -0.984471446 + -0.1041666667 w = -0.984471446 + -0.1041666667 Combine like terms: -0.984471446 + -0.1041666667 = -1.0886381127 w = -1.0886381127 Simplifying w = -1.0886381127Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.8803047793, -1.0886381127}
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