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Simplifying (2w + -2)(w + 4) = 7 Reorder the terms: (-2 + 2w)(w + 4) = 7 Reorder the terms: (-2 + 2w)(4 + w) = 7 Multiply (-2 + 2w) * (4 + w) (-2(4 + w) + 2w * (4 + w)) = 7 ((4 * -2 + w * -2) + 2w * (4 + w)) = 7 ((-8 + -2w) + 2w * (4 + w)) = 7 (-8 + -2w + (4 * 2w + w * 2w)) = 7 (-8 + -2w + (8w + 2w2)) = 7 Combine like terms: -2w + 8w = 6w (-8 + 6w + 2w2) = 7 Solving -8 + 6w + 2w2 = 7 Solving for variable 'w'. Reorder the terms: -8 + -7 + 6w + 2w2 = 7 + -7 Combine like terms: -8 + -7 = -15 -15 + 6w + 2w2 = 7 + -7 Combine like terms: 7 + -7 = 0 -15 + 6w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + 3w + w2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 3w + 7.5 + w2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 3w + w2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 3w + w2 = 0 + 7.5 3w + w2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 3w + w2 = 7.5 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 = 7.5 + 2.25 Reorder the terms: 2.25 + 3w + w2 = 7.5 + 2.25 Combine like terms: 7.5 + 2.25 = 9.75 2.25 + 3w + w2 = 9.75 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 9.75 Calculate the square root of the right side: 3.122498999 Break this problem into two subproblems by setting (w + 1.5) equal to 3.122498999 and -3.122498999.Subproblem 1
w + 1.5 = 3.122498999 Simplifying w + 1.5 = 3.122498999 Reorder the terms: 1.5 + w = 3.122498999 Solving 1.5 + w = 3.122498999 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 = 3.122498999 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 3.122498999 + -1.5 w = 3.122498999 + -1.5 Combine like terms: 3.122498999 + -1.5 = 1.622498999 w = 1.622498999 Simplifying w = 1.622498999Subproblem 2
w + 1.5 = -3.122498999 Simplifying w + 1.5 = -3.122498999 Reorder the terms: 1.5 + w = -3.122498999 Solving 1.5 + w = -3.122498999 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 = -3.122498999 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -3.122498999 + -1.5 w = -3.122498999 + -1.5 Combine like terms: -3.122498999 + -1.5 = -4.622498999 w = -4.622498999 Simplifying w = -4.622498999Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.622498999, -4.622498999}
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