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Simplifying (3w + 8)(w) = 154 Reorder the terms: (8 + 3w)(w) = 154 Reorder the terms for easier multiplication: w(8 + 3w) = 154 (8 * w + 3w * w) = 154 (8w + 3w2) = 154 Solving 8w + 3w2 = 154 Solving for variable 'w'. Reorder the terms: -154 + 8w + 3w2 = 154 + -154 Combine like terms: 154 + -154 = 0 -154 + 8w + 3w2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -51.33333333 + 2.666666667w + w2 = 0 Move the constant term to the right: Add '51.33333333' to each side of the equation. -51.33333333 + 2.666666667w + 51.33333333 + w2 = 0 + 51.33333333 Reorder the terms: -51.33333333 + 51.33333333 + 2.666666667w + w2 = 0 + 51.33333333 Combine like terms: -51.33333333 + 51.33333333 = 0.00000000 0.00000000 + 2.666666667w + w2 = 0 + 51.33333333 2.666666667w + w2 = 0 + 51.33333333 Combine like terms: 0 + 51.33333333 = 51.33333333 2.666666667w + w2 = 51.33333333 The w term is 2.666666667w. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667w + 1.777777780 + w2 = 51.33333333 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667w + w2 = 51.33333333 + 1.777777780 Combine like terms: 51.33333333 + 1.777777780 = 53.11111111 1.777777780 + 2.666666667w + w2 = 53.11111111 Factor a perfect square on the left side: (w + 1.333333334)(w + 1.333333334) = 53.11111111 Calculate the square root of the right side: 7.287737036 Break this problem into two subproblems by setting (w + 1.333333334) equal to 7.287737036 and -7.287737036.Subproblem 1
w + 1.333333334 = 7.287737036 Simplifying w + 1.333333334 = 7.287737036 Reorder the terms: 1.333333334 + w = 7.287737036 Solving 1.333333334 + w = 7.287737036 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + w = 7.287737036 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + w = 7.287737036 + -1.333333334 w = 7.287737036 + -1.333333334 Combine like terms: 7.287737036 + -1.333333334 = 5.954403702 w = 5.954403702 Simplifying w = 5.954403702Subproblem 2
w + 1.333333334 = -7.287737036 Simplifying w + 1.333333334 = -7.287737036 Reorder the terms: 1.333333334 + w = -7.287737036 Solving 1.333333334 + w = -7.287737036 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + w = -7.287737036 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + w = -7.287737036 + -1.333333334 w = -7.287737036 + -1.333333334 Combine like terms: -7.287737036 + -1.333333334 = -8.62107037 w = -8.62107037 Simplifying w = -8.62107037Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.954403702, -8.62107037}
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