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Simplifying 3w2 + 11w = 77 Reorder the terms: 11w + 3w2 = 77 Solving 11w + 3w2 = 77 Solving for variable 'w'. Reorder the terms: -77 + 11w + 3w2 = 77 + -77 Combine like terms: 77 + -77 = 0 -77 + 11w + 3w2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -25.66666667 + 3.666666667w + w2 = 0 Move the constant term to the right: Add '25.66666667' to each side of the equation. -25.66666667 + 3.666666667w + 25.66666667 + w2 = 0 + 25.66666667 Reorder the terms: -25.66666667 + 25.66666667 + 3.666666667w + w2 = 0 + 25.66666667 Combine like terms: -25.66666667 + 25.66666667 = 0.00000000 0.00000000 + 3.666666667w + w2 = 0 + 25.66666667 3.666666667w + w2 = 0 + 25.66666667 Combine like terms: 0 + 25.66666667 = 25.66666667 3.666666667w + w2 = 25.66666667 The w term is 3.666666667w. Take half its coefficient (1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. 3.666666667w + 3.361111114 + w2 = 25.66666667 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667w + w2 = 25.66666667 + 3.361111114 Combine like terms: 25.66666667 + 3.361111114 = 29.027777784 3.361111114 + 3.666666667w + w2 = 29.027777784 Factor a perfect square on the left side: (w + 1.833333334)(w + 1.833333334) = 29.027777784 Calculate the square root of the right side: 5.387743292 Break this problem into two subproblems by setting (w + 1.833333334) equal to 5.387743292 and -5.387743292.Subproblem 1
w + 1.833333334 = 5.387743292 Simplifying w + 1.833333334 = 5.387743292 Reorder the terms: 1.833333334 + w = 5.387743292 Solving 1.833333334 + w = 5.387743292 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + w = 5.387743292 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + w = 5.387743292 + -1.833333334 w = 5.387743292 + -1.833333334 Combine like terms: 5.387743292 + -1.833333334 = 3.554409958 w = 3.554409958 Simplifying w = 3.554409958Subproblem 2
w + 1.833333334 = -5.387743292 Simplifying w + 1.833333334 = -5.387743292 Reorder the terms: 1.833333334 + w = -5.387743292 Solving 1.833333334 + w = -5.387743292 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + w = -5.387743292 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + w = -5.387743292 + -1.833333334 w = -5.387743292 + -1.833333334 Combine like terms: -5.387743292 + -1.833333334 = -7.221076626 w = -7.221076626 Simplifying w = -7.221076626Solution
The solution to the problem is based on the solutions from the subproblems. w = {3.554409958, -7.221076626}
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