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Simplifying (2w2) + w + -68 = 0 Reorder the terms: -68 + w + (2w2) = 0 Solving -68 + w + (2w2) = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -34 + 0.5w + w2 = 0 Move the constant term to the right: Add '34' to each side of the equation. -34 + 0.5w + 34 + w2 = 0 + 34 Reorder the terms: -34 + 34 + 0.5w + w2 = 0 + 34 Combine like terms: -34 + 34 = 0 0 + 0.5w + w2 = 0 + 34 0.5w + w2 = 0 + 34 Combine like terms: 0 + 34 = 34 0.5w + w2 = 34 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. 0.5w + 0.25 + w2 = 34 + 0.25 Reorder the terms: 0.25 + 0.5w + w2 = 34 + 0.25 Combine like terms: 34 + 0.25 = 34.25 0.25 + 0.5w + w2 = 34.25 Factor a perfect square on the left side: ((w) + 0.5)((w) + 0.5) = 34.25 Calculate the square root of the right side: 5.852349955 Break this problem into two subproblems by setting ((w) + 0.5) equal to 5.852349955 and -5.852349955.Subproblem 1
(w) + 0.5 = 5.852349955 Simplifying (w) + 0.5 = 5.852349955 w + 0.5 = 5.852349955 Reorder the terms: 0.5 + w = 5.852349955 Solving 0.5 + w = 5.852349955 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 = 5.852349955 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 5.852349955 + -0.5 w = 5.852349955 + -0.5 Combine like terms: 5.852349955 + -0.5 = 5.352349955 w = 5.352349955 Simplifying w = 5.352349955Subproblem 2
(w) + 0.5 = -5.852349955 Simplifying (w) + 0.5 = -5.852349955 w + 0.5 = -5.852349955 Reorder the terms: 0.5 + w = -5.852349955 Solving 0.5 + w = -5.852349955 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 = -5.852349955 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -5.852349955 + -0.5 w = -5.852349955 + -0.5 Combine like terms: -5.852349955 + -0.5 = -6.352349955 w = -6.352349955 Simplifying w = -6.352349955Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.352349955, -6.352349955}
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