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Simplifying -1w2 + 23w + 14 = 0 Reorder the terms: 14 + 23w + -1w2 = 0 Solving 14 + 23w + -1w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -14 + -23w + w2 = 0 Move the constant term to the right: Add '14' to each side of the equation. -14 + -23w + 14 + w2 = 0 + 14 Reorder the terms: -14 + 14 + -23w + w2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + -23w + w2 = 0 + 14 -23w + w2 = 0 + 14 Combine like terms: 0 + 14 = 14 -23w + w2 = 14 The w term is -23w. Take half its coefficient (-11.5). Square it (132.25) and add it to both sides. Add '132.25' to each side of the equation. -23w + 132.25 + w2 = 14 + 132.25 Reorder the terms: 132.25 + -23w + w2 = 14 + 132.25 Combine like terms: 14 + 132.25 = 146.25 132.25 + -23w + w2 = 146.25 Factor a perfect square on the left side: (w + -11.5)(w + -11.5) = 146.25 Calculate the square root of the right side: 12.093386622 Break this problem into two subproblems by setting (w + -11.5) equal to 12.093386622 and -12.093386622.Subproblem 1
w + -11.5 = 12.093386622 Simplifying w + -11.5 = 12.093386622 Reorder the terms: -11.5 + w = 12.093386622 Solving -11.5 + w = 12.093386622 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + w = 12.093386622 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + w = 12.093386622 + 11.5 w = 12.093386622 + 11.5 Combine like terms: 12.093386622 + 11.5 = 23.593386622 w = 23.593386622 Simplifying w = 23.593386622Subproblem 2
w + -11.5 = -12.093386622 Simplifying w + -11.5 = -12.093386622 Reorder the terms: -11.5 + w = -12.093386622 Solving -11.5 + w = -12.093386622 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + w = -12.093386622 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + w = -12.093386622 + 11.5 w = -12.093386622 + 11.5 Combine like terms: -12.093386622 + 11.5 = -0.593386622 w = -0.593386622 Simplifying w = -0.593386622Solution
The solution to the problem is based on the solutions from the subproblems. w = {23.593386622, -0.593386622}
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