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Simplifying 0.6w + 2w2 = 4.6 Solving 0.6w + 2w2 = 4.6 Solving for variable 'w'. Reorder the terms: -4.6 + 0.6w + 2w2 = 4.6 + -4.6 Combine like terms: 4.6 + -4.6 = 0.0 -4.6 + 0.6w + 2w2 = 0.0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2.3 + 0.3w + w2 = 0.0 Move the constant term to the right: Add '2.3' to each side of the equation. -2.3 + 0.3w + 2.3 + w2 = 0.0 + 2.3 Reorder the terms: -2.3 + 2.3 + 0.3w + w2 = 0.0 + 2.3 Combine like terms: -2.3 + 2.3 = 0.0 0.0 + 0.3w + w2 = 0.0 + 2.3 0.3w + w2 = 0.0 + 2.3 Combine like terms: 0.0 + 2.3 = 2.3 0.3w + w2 = 2.3 The w term is 0.3w. Take half its coefficient (0.15). Square it (0.0225) and add it to both sides. Add '0.0225' to each side of the equation. 0.3w + 0.0225 + w2 = 2.3 + 0.0225 Reorder the terms: 0.0225 + 0.3w + w2 = 2.3 + 0.0225 Combine like terms: 2.3 + 0.0225 = 2.3225 0.0225 + 0.3w + w2 = 2.3225 Factor a perfect square on the left side: (w + 0.15)(w + 0.15) = 2.3225 Calculate the square root of the right side: 1.523975065 Break this problem into two subproblems by setting (w + 0.15) equal to 1.523975065 and -1.523975065.Subproblem 1
w + 0.15 = 1.523975065 Simplifying w + 0.15 = 1.523975065 Reorder the terms: 0.15 + w = 1.523975065 Solving 0.15 + w = 1.523975065 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + w = 1.523975065 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + w = 1.523975065 + -0.15 w = 1.523975065 + -0.15 Combine like terms: 1.523975065 + -0.15 = 1.373975065 w = 1.373975065 Simplifying w = 1.373975065Subproblem 2
w + 0.15 = -1.523975065 Simplifying w + 0.15 = -1.523975065 Reorder the terms: 0.15 + w = -1.523975065 Solving 0.15 + w = -1.523975065 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + w = -1.523975065 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + w = -1.523975065 + -0.15 w = -1.523975065 + -0.15 Combine like terms: -1.523975065 + -0.15 = -1.673975065 w = -1.673975065 Simplifying w = -1.673975065Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.373975065, -1.673975065}
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