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Simplifying 3k3 + 30k2 + -24k = 0 Reorder the terms: -24k + 30k2 + 3k3 = 0 Solving -24k + 30k2 + 3k3 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '3k'. 3k(-8 + 10k + k2) = 0 Ignore the factor 3.Subproblem 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 '(-8 + 10k + k2)' equal to zero and attempt to solve: Simplifying -8 + 10k + k2 = 0 Solving -8 + 10k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '8' to each side of the equation. -8 + 10k + 8 + k2 = 0 + 8 Reorder the terms: -8 + 8 + 10k + k2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 10k + k2 = 0 + 8 10k + k2 = 0 + 8 Combine like terms: 0 + 8 = 8 10k + k2 = 8 The k term is 10k. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10k + 25 + k2 = 8 + 25 Reorder the terms: 25 + 10k + k2 = 8 + 25 Combine like terms: 8 + 25 = 33 25 + 10k + k2 = 33 Factor a perfect square on the left side: (k + 5)(k + 5) = 33 Calculate the square root of the right side: 5.744562647 Break this problem into two subproblems by setting (k + 5) equal to 5.744562647 and -5.744562647.Subproblem 1
k + 5 = 5.744562647 Simplifying k + 5 = 5.744562647 Reorder the terms: 5 + k = 5.744562647 Solving 5 + k = 5.744562647 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + k = 5.744562647 + -5 Combine like terms: 5 + -5 = 0 0 + k = 5.744562647 + -5 k = 5.744562647 + -5 Combine like terms: 5.744562647 + -5 = 0.744562647 k = 0.744562647 Simplifying k = 0.744562647Subproblem 2
k + 5 = -5.744562647 Simplifying k + 5 = -5.744562647 Reorder the terms: 5 + k = -5.744562647 Solving 5 + k = -5.744562647 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + k = -5.744562647 + -5 Combine like terms: 5 + -5 = 0 0 + k = -5.744562647 + -5 k = -5.744562647 + -5 Combine like terms: -5.744562647 + -5 = -10.744562647 k = -10.744562647 Simplifying k = -10.744562647Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.744562647, -10.744562647}Solution
k = {0, 0.744562647, -10.744562647}
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