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Simplifying w3 + -1w2 + -40w = 0 Reorder the terms: -40w + -1w2 + w3 = 0 Solving -40w + -1w2 + w3 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), 'w'. w(-40 + -1w + w2) = 0Subproblem 1
Set the factor 'w' equal to zero and attempt to solve: Simplifying w = 0 Solving w = 0 Move all terms containing w to the left, all other terms to the right. Simplifying w = 0Subproblem 2
Set the factor '(-40 + -1w + w2)' equal to zero and attempt to solve: Simplifying -40 + -1w + w2 = 0 Solving -40 + -1w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '40' to each side of the equation. -40 + -1w + 40 + w2 = 0 + 40 Reorder the terms: -40 + 40 + -1w + w2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + -1w + w2 = 0 + 40 -1w + w2 = 0 + 40 Combine like terms: 0 + 40 = 40 -1w + w2 = 40 The w term is -1w. 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. -1w + 0.25 + w2 = 40 + 0.25 Reorder the terms: 0.25 + -1w + w2 = 40 + 0.25 Combine like terms: 40 + 0.25 = 40.25 0.25 + -1w + w2 = 40.25 Factor a perfect square on the left side: (w + -0.5)(w + -0.5) = 40.25 Calculate the square root of the right side: 6.34428877 Break this problem into two subproblems by setting (w + -0.5) equal to 6.34428877 and -6.34428877.Subproblem 1
w + -0.5 = 6.34428877 Simplifying w + -0.5 = 6.34428877 Reorder the terms: -0.5 + w = 6.34428877 Solving -0.5 + w = 6.34428877 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 = 6.34428877 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + w = 6.34428877 + 0.5 w = 6.34428877 + 0.5 Combine like terms: 6.34428877 + 0.5 = 6.84428877 w = 6.84428877 Simplifying w = 6.84428877Subproblem 2
w + -0.5 = -6.34428877 Simplifying w + -0.5 = -6.34428877 Reorder the terms: -0.5 + w = -6.34428877 Solving -0.5 + w = -6.34428877 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 = -6.34428877 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + w = -6.34428877 + 0.5 w = -6.34428877 + 0.5 Combine like terms: -6.34428877 + 0.5 = -5.84428877 w = -5.84428877 Simplifying w = -5.84428877Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.84428877, -5.84428877}Solution
w = {0, 6.84428877, -5.84428877}
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