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Simplifying k3 + -6k2 + -21k = 0 Reorder the terms: -21k + -6k2 + k3 = 0 Solving -21k + -6k2 + k3 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), 'k'. k(-21 + -6k + 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 '(-21 + -6k + k2)' equal to zero and attempt to solve: Simplifying -21 + -6k + k2 = 0 Solving -21 + -6k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '21' to each side of the equation. -21 + -6k + 21 + k2 = 0 + 21 Reorder the terms: -21 + 21 + -6k + k2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + -6k + k2 = 0 + 21 -6k + k2 = 0 + 21 Combine like terms: 0 + 21 = 21 -6k + k2 = 21 The k term is -6k. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6k + 9 + k2 = 21 + 9 Reorder the terms: 9 + -6k + k2 = 21 + 9 Combine like terms: 21 + 9 = 30 9 + -6k + k2 = 30 Factor a perfect square on the left side: (k + -3)(k + -3) = 30 Calculate the square root of the right side: 5.477225575 Break this problem into two subproblems by setting (k + -3) equal to 5.477225575 and -5.477225575.Subproblem 1
k + -3 = 5.477225575 Simplifying k + -3 = 5.477225575 Reorder the terms: -3 + k = 5.477225575 Solving -3 + k = 5.477225575 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + k = 5.477225575 + 3 Combine like terms: -3 + 3 = 0 0 + k = 5.477225575 + 3 k = 5.477225575 + 3 Combine like terms: 5.477225575 + 3 = 8.477225575 k = 8.477225575 Simplifying k = 8.477225575Subproblem 2
k + -3 = -5.477225575 Simplifying k + -3 = -5.477225575 Reorder the terms: -3 + k = -5.477225575 Solving -3 + k = -5.477225575 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + k = -5.477225575 + 3 Combine like terms: -3 + 3 = 0 0 + k = -5.477225575 + 3 k = -5.477225575 + 3 Combine like terms: -5.477225575 + 3 = -2.477225575 k = -2.477225575 Simplifying k = -2.477225575Solution
The solution to the problem is based on the solutions from the subproblems. k = {8.477225575, -2.477225575}Solution
k = {0, 8.477225575, -2.477225575}
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