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Simplifying k3 + -15k2 + 5k = 0 Reorder the terms: 5k + -15k2 + k3 = 0 Solving 5k + -15k2 + k3 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), 'k'. k(5 + -15k + k2) = 0Subproblem 1
Set the factor 'k' equal to zero and attempt to solve: Simplifying k = 0 Solving k = 0 Move all terms containing k to the left, all other terms to the right. Simplifying k = 0Subproblem 2
Set the factor '(5 + -15k + k2)' equal to zero and attempt to solve: Simplifying 5 + -15k + k2 = 0 Solving 5 + -15k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + -15k + -5 + k2 = 0 + -5 Reorder the terms: 5 + -5 + -15k + k2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + -15k + k2 = 0 + -5 -15k + k2 = 0 + -5 Combine like terms: 0 + -5 = -5 -15k + k2 = -5 The k term is -15k. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15k + 56.25 + k2 = -5 + 56.25 Reorder the terms: 56.25 + -15k + k2 = -5 + 56.25 Combine like terms: -5 + 56.25 = 51.25 56.25 + -15k + k2 = 51.25 Factor a perfect square on the left side: (k + -7.5)(k + -7.5) = 51.25 Calculate the square root of the right side: 7.158910532 Break this problem into two subproblems by setting (k + -7.5) equal to 7.158910532 and -7.158910532.Subproblem 1
k + -7.5 = 7.158910532 Simplifying k + -7.5 = 7.158910532 Reorder the terms: -7.5 + k = 7.158910532 Solving -7.5 + k = 7.158910532 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + k = 7.158910532 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + k = 7.158910532 + 7.5 k = 7.158910532 + 7.5 Combine like terms: 7.158910532 + 7.5 = 14.658910532 k = 14.658910532 Simplifying k = 14.658910532Subproblem 2
k + -7.5 = -7.158910532 Simplifying k + -7.5 = -7.158910532 Reorder the terms: -7.5 + k = -7.158910532 Solving -7.5 + k = -7.158910532 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + k = -7.158910532 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + k = -7.158910532 + 7.5 k = -7.158910532 + 7.5 Combine like terms: -7.158910532 + 7.5 = 0.341089468 k = 0.341089468 Simplifying k = 0.341089468Solution
The solution to the problem is based on the solutions from the subproblems. k = {14.658910532, 0.341089468}Solution
k = {0, 14.658910532, 0.341089468}
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