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Simplifying 4w2 + w + 1 = 0 Reorder the terms: 1 + w + 4w2 = 0 Solving 1 + w + 4w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + 0.25w + w2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + 0.25w + -0.25 + w2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + 0.25w + w2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + 0.25w + w2 = 0 + -0.25 0.25w + w2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 0.25w + w2 = -0.25 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. 0.25w + 0.25 + w2 = -0.25 + 0.25 Reorder the terms: 0.25 + 0.25w + w2 = -0.25 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.25 + 0.25w + w2 = 0.00 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 0.00 Calculate the square root of the right side: 0.00 Break this problem into two subproblems by setting (w + 0.5) equal to 0.00 and 0.00.Subproblem 1
w + 0.5 = 0.00 Simplifying w + 0.5 = 0.00 Reorder the terms: 0.5 + w = 0.00 Solving 0.5 + w = 0.00 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 = 0.00 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 0.00 + -0.5 w = 0.00 + -0.5 Combine like terms: 0.00 + -0.5 = -0.5 w = -0.5 Simplifying w = -0.5Subproblem 2
w + 0.5 = 0.00 Simplifying w + 0.5 = 0.00 Reorder the terms: 0.5 + w = 0.00 Solving 0.5 + w = 0.00 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 = 0.00 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 0.00 + -0.5 w = 0.00 + -0.5 Combine like terms: 0.00 + -0.5 = -0.5 w = -0.5 Simplifying w = -0.5Solution
The solution to the problem is based on the solutions from the subproblems. w = {-0.5, -0.5}
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