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Simplifying 3w2 + w + -68 = 0 Reorder the terms: -68 + w + 3w2 = 0 Solving -68 + w + 3w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -22.66666667 + 0.3333333333w + w2 = 0 Move the constant term to the right: Add '22.66666667' to each side of the equation. -22.66666667 + 0.3333333333w + 22.66666667 + w2 = 0 + 22.66666667 Reorder the terms: -22.66666667 + 22.66666667 + 0.3333333333w + w2 = 0 + 22.66666667 Combine like terms: -22.66666667 + 22.66666667 = 0.00000000 0.00000000 + 0.3333333333w + w2 = 0 + 22.66666667 0.3333333333w + w2 = 0 + 22.66666667 Combine like terms: 0 + 22.66666667 = 22.66666667 0.3333333333w + w2 = 22.66666667 The w term is w. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.3333333333w + 0.25 + w2 = 22.66666667 + 0.25 Reorder the terms: 0.25 + 0.3333333333w + w2 = 22.66666667 + 0.25 Combine like terms: 22.66666667 + 0.25 = 22.91666667 0.25 + 0.3333333333w + w2 = 22.91666667 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 22.91666667 Calculate the square root of the right side: 4.787135539 Break this problem into two subproblems by setting (w + 0.5) equal to 4.787135539 and -4.787135539.Subproblem 1
w + 0.5 = 4.787135539 Simplifying w + 0.5 = 4.787135539 Reorder the terms: 0.5 + w = 4.787135539 Solving 0.5 + w = 4.787135539 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = 4.787135539 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 4.787135539 + -0.5 w = 4.787135539 + -0.5 Combine like terms: 4.787135539 + -0.5 = 4.287135539 w = 4.287135539 Simplifying w = 4.287135539Subproblem 2
w + 0.5 = -4.787135539 Simplifying w + 0.5 = -4.787135539 Reorder the terms: 0.5 + w = -4.787135539 Solving 0.5 + w = -4.787135539 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = -4.787135539 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -4.787135539 + -0.5 w = -4.787135539 + -0.5 Combine like terms: -4.787135539 + -0.5 = -5.287135539 w = -5.287135539 Simplifying w = -5.287135539Solution
The solution to the problem is based on the solutions from the subproblems. w = {4.287135539, -5.287135539}
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