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Simplifying 5w2 + -5w + -115 = 0 Reorder the terms: -115 + -5w + 5w2 = 0 Solving -115 + -5w + 5w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '5'. 5(-23 + -1w + w2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-23 + -1w + w2)' equal to zero and attempt to solve: Simplifying -23 + -1w + w2 = 0 Solving -23 + -1w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '23' to each side of the equation. -23 + -1w + 23 + w2 = 0 + 23 Reorder the terms: -23 + 23 + -1w + w2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + -1w + w2 = 0 + 23 -1w + w2 = 0 + 23 Combine like terms: 0 + 23 = 23 -1w + w2 = 23 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 = 23 + 0.25 Reorder the terms: 0.25 + -1w + w2 = 23 + 0.25 Combine like terms: 23 + 0.25 = 23.25 0.25 + -1w + w2 = 23.25 Factor a perfect square on the left side: (w + -0.5)(w + -0.5) = 23.25 Calculate the square root of the right side: 4.82182538 Break this problem into two subproblems by setting (w + -0.5) equal to 4.82182538 and -4.82182538.Subproblem 1
w + -0.5 = 4.82182538 Simplifying w + -0.5 = 4.82182538 Reorder the terms: -0.5 + w = 4.82182538 Solving -0.5 + w = 4.82182538 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 = 4.82182538 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + w = 4.82182538 + 0.5 w = 4.82182538 + 0.5 Combine like terms: 4.82182538 + 0.5 = 5.32182538 w = 5.32182538 Simplifying w = 5.32182538Subproblem 2
w + -0.5 = -4.82182538 Simplifying w + -0.5 = -4.82182538 Reorder the terms: -0.5 + w = -4.82182538 Solving -0.5 + w = -4.82182538 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 = -4.82182538 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + w = -4.82182538 + 0.5 w = -4.82182538 + 0.5 Combine like terms: -4.82182538 + 0.5 = -4.32182538 w = -4.32182538 Simplifying w = -4.32182538Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.32182538, -4.32182538}Solution
w = {5.32182538, -4.32182538}
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