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Simplifying 3.5 = w2 + w Reorder the terms: 3.5 = w + w2 Solving 3.5 = w + w2 Solving for variable 'w'. Reorder the terms: 3.5 + -1w + -1w2 = w + -1w + w2 + -1w2 Combine like terms: w + -1w = 0 3.5 + -1w + -1w2 = 0 + w2 + -1w2 3.5 + -1w + -1w2 = w2 + -1w2 Combine like terms: w2 + -1w2 = 0 3.5 + -1w + -1w2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -3.5 + w + w2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + w + 3.5 + w2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + w + w2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + w + w2 = 0 + 3.5 w + w2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 w + w2 = 3.5 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. + 0.25 + w2 = 3.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + w2 = 3.5 + 0.25 Combine like terms: 3.5 + 0.25 = 3.75 1.25 + w2 = 3.75 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (w + 0.5) equal to 1.936491673 and -1.936491673.Subproblem 1
w + 0.5 = 1.936491673 Simplifying w + 0.5 = 1.936491673 Reorder the terms: 0.5 + w = 1.936491673 Solving 0.5 + w = 1.936491673 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 = 1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 1.936491673 + -0.5 w = 1.936491673 + -0.5 Combine like terms: 1.936491673 + -0.5 = 1.436491673 w = 1.436491673 Simplifying w = 1.436491673Subproblem 2
w + 0.5 = -1.936491673 Simplifying w + 0.5 = -1.936491673 Reorder the terms: 0.5 + w = -1.936491673 Solving 0.5 + w = -1.936491673 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 = -1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -1.936491673 + -0.5 w = -1.936491673 + -0.5 Combine like terms: -1.936491673 + -0.5 = -2.436491673 w = -2.436491673 Simplifying w = -2.436491673Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.436491673, -2.436491673}
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