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Simplifying 24 + -1w * w = 3w + -8 Multiply w * w 24 + -1w2 = 3w + -8 Reorder the terms: 24 + -1w2 = -8 + 3w Solving 24 + -1w2 = -8 + 3w Solving for variable 'w'. Reorder the terms: 24 + 8 + -3w + -1w2 = -8 + 3w + 8 + -3w Combine like terms: 24 + 8 = 32 32 + -3w + -1w2 = -8 + 3w + 8 + -3w Reorder the terms: 32 + -3w + -1w2 = -8 + 8 + 3w + -3w Combine like terms: -8 + 8 = 0 32 + -3w + -1w2 = 0 + 3w + -3w 32 + -3w + -1w2 = 3w + -3w Combine like terms: 3w + -3w = 0 32 + -3w + -1w2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -32 + 3w + w2 = 0 Move the constant term to the right: Add '32' to each side of the equation. -32 + 3w + 32 + w2 = 0 + 32 Reorder the terms: -32 + 32 + 3w + w2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + 3w + w2 = 0 + 32 3w + w2 = 0 + 32 Combine like terms: 0 + 32 = 32 3w + w2 = 32 The w term is 3w. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3w + 2.25 + w2 = 32 + 2.25 Reorder the terms: 2.25 + 3w + w2 = 32 + 2.25 Combine like terms: 32 + 2.25 = 34.25 2.25 + 3w + w2 = 34.25 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 34.25 Calculate the square root of the right side: 5.852349955 Break this problem into two subproblems by setting (w + 1.5) equal to 5.852349955 and -5.852349955.Subproblem 1
w + 1.5 = 5.852349955 Simplifying w + 1.5 = 5.852349955 Reorder the terms: 1.5 + w = 5.852349955 Solving 1.5 + w = 5.852349955 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 5.852349955 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 5.852349955 + -1.5 w = 5.852349955 + -1.5 Combine like terms: 5.852349955 + -1.5 = 4.352349955 w = 4.352349955 Simplifying w = 4.352349955Subproblem 2
w + 1.5 = -5.852349955 Simplifying w + 1.5 = -5.852349955 Reorder the terms: 1.5 + w = -5.852349955 Solving 1.5 + w = -5.852349955 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -5.852349955 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -5.852349955 + -1.5 w = -5.852349955 + -1.5 Combine like terms: -5.852349955 + -1.5 = -7.352349955 w = -7.352349955 Simplifying w = -7.352349955Solution
The solution to the problem is based on the solutions from the subproblems. w = {4.352349955, -7.352349955}
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