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Simplifying w3 + -1w(w2 + 2w + -1) + 2 = 0 Reorder the terms: w3 + -1w(-1 + 2w + w2) + 2 = 0 w3 + (-1 * -1w + 2w * -1w + w2 * -1w) + 2 = 0 w3 + (1w + -2w2 + -1w3) + 2 = 0 Reorder the terms: 2 + 1w + -2w2 + w3 + -1w3 = 0 Combine like terms: w3 + -1w3 = 0 2 + 1w + -2w2 + 0 = 0 2 + 1w + -2w2 = 0 Solving 2 + 1w + -2w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1 + -0.5w + w2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.5w + 1 + w2 = 0 + 1 Reorder the terms: -1 + 1 + -0.5w + w2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.5w + w2 = 0 + 1 -0.5w + w2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.5w + w2 = 1 The w term is -0.5w. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5w + 0.0625 + w2 = 1 + 0.0625 Reorder the terms: 0.0625 + -0.5w + w2 = 1 + 0.0625 Combine like terms: 1 + 0.0625 = 1.0625 0.0625 + -0.5w + w2 = 1.0625 Factor a perfect square on the left side: (w + -0.25)(w + -0.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (w + -0.25) equal to 1.030776406 and -1.030776406.Subproblem 1
w + -0.25 = 1.030776406 Simplifying w + -0.25 = 1.030776406 Reorder the terms: -0.25 + w = 1.030776406 Solving -0.25 + w = 1.030776406 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + w = 1.030776406 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + w = 1.030776406 + 0.25 w = 1.030776406 + 0.25 Combine like terms: 1.030776406 + 0.25 = 1.280776406 w = 1.280776406 Simplifying w = 1.280776406Subproblem 2
w + -0.25 = -1.030776406 Simplifying w + -0.25 = -1.030776406 Reorder the terms: -0.25 + w = -1.030776406 Solving -0.25 + w = -1.030776406 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + w = -1.030776406 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + w = -1.030776406 + 0.25 w = -1.030776406 + 0.25 Combine like terms: -1.030776406 + 0.25 = -0.780776406 w = -0.780776406 Simplifying w = -0.780776406Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.280776406, -0.780776406}
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