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Simplifying 2w2 + 21w + 52 = 69 Reorder the terms: 52 + 21w + 2w2 = 69 Solving 52 + 21w + 2w2 = 69 Solving for variable 'w'. Reorder the terms: 52 + -69 + 21w + 2w2 = 69 + -69 Combine like terms: 52 + -69 = -17 -17 + 21w + 2w2 = 69 + -69 Combine like terms: 69 + -69 = 0 -17 + 21w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -8.5 + 10.5w + w2 = 0 Move the constant term to the right: Add '8.5' to each side of the equation. -8.5 + 10.5w + 8.5 + w2 = 0 + 8.5 Reorder the terms: -8.5 + 8.5 + 10.5w + w2 = 0 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + 10.5w + w2 = 0 + 8.5 10.5w + w2 = 0 + 8.5 Combine like terms: 0 + 8.5 = 8.5 10.5w + w2 = 8.5 The w term is 10.5w. Take half its coefficient (5.25). Square it (27.5625) and add it to both sides. Add '27.5625' to each side of the equation. 10.5w + 27.5625 + w2 = 8.5 + 27.5625 Reorder the terms: 27.5625 + 10.5w + w2 = 8.5 + 27.5625 Combine like terms: 8.5 + 27.5625 = 36.0625 27.5625 + 10.5w + w2 = 36.0625 Factor a perfect square on the left side: (w + 5.25)(w + 5.25) = 36.0625 Calculate the square root of the right side: 6.005206075 Break this problem into two subproblems by setting (w + 5.25) equal to 6.005206075 and -6.005206075.Subproblem 1
w + 5.25 = 6.005206075 Simplifying w + 5.25 = 6.005206075 Reorder the terms: 5.25 + w = 6.005206075 Solving 5.25 + w = 6.005206075 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + w = 6.005206075 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + w = 6.005206075 + -5.25 w = 6.005206075 + -5.25 Combine like terms: 6.005206075 + -5.25 = 0.755206075 w = 0.755206075 Simplifying w = 0.755206075Subproblem 2
w + 5.25 = -6.005206075 Simplifying w + 5.25 = -6.005206075 Reorder the terms: 5.25 + w = -6.005206075 Solving 5.25 + w = -6.005206075 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + w = -6.005206075 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + w = -6.005206075 + -5.25 w = -6.005206075 + -5.25 Combine like terms: -6.005206075 + -5.25 = -11.255206075 w = -11.255206075 Simplifying w = -11.255206075Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.755206075, -11.255206075}
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