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Simplifying (-1w2 + 3w + -6) + (5w2 + 2w + 4) = 0 Reorder the terms: (-6 + 3w + -1w2) + (5w2 + 2w + 4) = 0 Remove parenthesis around (-6 + 3w + -1w2) -6 + 3w + -1w2 + (5w2 + 2w + 4) = 0 Reorder the terms: -6 + 3w + -1w2 + (4 + 2w + 5w2) = 0 Remove parenthesis around (4 + 2w + 5w2) -6 + 3w + -1w2 + 4 + 2w + 5w2 = 0 Reorder the terms: -6 + 4 + 3w + 2w + -1w2 + 5w2 = 0 Combine like terms: -6 + 4 = -2 -2 + 3w + 2w + -1w2 + 5w2 = 0 Combine like terms: 3w + 2w = 5w -2 + 5w + -1w2 + 5w2 = 0 Combine like terms: -1w2 + 5w2 = 4w2 -2 + 5w + 4w2 = 0 Solving -2 + 5w + 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'. -0.5 + 1.25w + w2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 1.25w + 0.5 + w2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 1.25w + w2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 1.25w + w2 = 0 + 0.5 1.25w + w2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 1.25w + w2 = 0.5 The w term is 1.25w. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25w + 0.390625 + w2 = 0.5 + 0.390625 Reorder the terms: 0.390625 + 1.25w + w2 = 0.5 + 0.390625 Combine like terms: 0.5 + 0.390625 = 0.890625 0.390625 + 1.25w + w2 = 0.890625 Factor a perfect square on the left side: (w + 0.625)(w + 0.625) = 0.890625 Calculate the square root of the right side: 0.943729304 Break this problem into two subproblems by setting (w + 0.625) equal to 0.943729304 and -0.943729304.Subproblem 1
w + 0.625 = 0.943729304 Simplifying w + 0.625 = 0.943729304 Reorder the terms: 0.625 + w = 0.943729304 Solving 0.625 + w = 0.943729304 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + w = 0.943729304 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + w = 0.943729304 + -0.625 w = 0.943729304 + -0.625 Combine like terms: 0.943729304 + -0.625 = 0.318729304 w = 0.318729304 Simplifying w = 0.318729304Subproblem 2
w + 0.625 = -0.943729304 Simplifying w + 0.625 = -0.943729304 Reorder the terms: 0.625 + w = -0.943729304 Solving 0.625 + w = -0.943729304 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + w = -0.943729304 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + w = -0.943729304 + -0.625 w = -0.943729304 + -0.625 Combine like terms: -0.943729304 + -0.625 = -1.568729304 w = -1.568729304 Simplifying w = -1.568729304Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.318729304, -1.568729304}
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