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Simplifying 4w2 + 12w + 8 = 81 Reorder the terms: 8 + 12w + 4w2 = 81 Solving 8 + 12w + 4w2 = 81 Solving for variable 'w'. Reorder the terms: 8 + -81 + 12w + 4w2 = 81 + -81 Combine like terms: 8 + -81 = -73 -73 + 12w + 4w2 = 81 + -81 Combine like terms: 81 + -81 = 0 -73 + 12w + 4w2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -18.25 + 3w + w2 = 0 Move the constant term to the right: Add '18.25' to each side of the equation. -18.25 + 3w + 18.25 + w2 = 0 + 18.25 Reorder the terms: -18.25 + 18.25 + 3w + w2 = 0 + 18.25 Combine like terms: -18.25 + 18.25 = 0.00 0.00 + 3w + w2 = 0 + 18.25 3w + w2 = 0 + 18.25 Combine like terms: 0 + 18.25 = 18.25 3w + w2 = 18.25 The w term is 3w. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3w + 2.25 + w2 = 18.25 + 2.25 Reorder the terms: 2.25 + 3w + w2 = 18.25 + 2.25 Combine like terms: 18.25 + 2.25 = 20.5 2.25 + 3w + w2 = 20.5 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 20.5 Calculate the square root of the right side: 4.527692569 Break this problem into two subproblems by setting (w + 1.5) equal to 4.527692569 and -4.527692569.Subproblem 1
w + 1.5 = 4.527692569 Simplifying w + 1.5 = 4.527692569 Reorder the terms: 1.5 + w = 4.527692569 Solving 1.5 + w = 4.527692569 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 4.527692569 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 4.527692569 + -1.5 w = 4.527692569 + -1.5 Combine like terms: 4.527692569 + -1.5 = 3.027692569 w = 3.027692569 Simplifying w = 3.027692569Subproblem 2
w + 1.5 = -4.527692569 Simplifying w + 1.5 = -4.527692569 Reorder the terms: 1.5 + w = -4.527692569 Solving 1.5 + w = -4.527692569 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -4.527692569 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -4.527692569 + -1.5 w = -4.527692569 + -1.5 Combine like terms: -4.527692569 + -1.5 = -6.027692569 w = -6.027692569 Simplifying w = -6.027692569Solution
The solution to the problem is based on the solutions from the subproblems. w = {3.027692569, -6.027692569}
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