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Simplifying 2k3 + -36k2 = 162k Reorder the terms: -36k2 + 2k3 = 162k Solving -36k2 + 2k3 = 162k Solving for variable 'k'. Reorder the terms: -162k + -36k2 + 2k3 = 162k + -162k Combine like terms: 162k + -162k = 0 -162k + -36k2 + 2k3 = 0 Factor out the Greatest Common Factor (GCF), '2k'. 2k(-81 + -18k + k2) = 0 Ignore the factor 2.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 '(-81 + -18k + k2)' equal to zero and attempt to solve: Simplifying -81 + -18k + k2 = 0 Solving -81 + -18k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '81' to each side of the equation. -81 + -18k + 81 + k2 = 0 + 81 Reorder the terms: -81 + 81 + -18k + k2 = 0 + 81 Combine like terms: -81 + 81 = 0 0 + -18k + k2 = 0 + 81 -18k + k2 = 0 + 81 Combine like terms: 0 + 81 = 81 -18k + k2 = 81 The k term is -18k. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18k + 81 + k2 = 81 + 81 Reorder the terms: 81 + -18k + k2 = 81 + 81 Combine like terms: 81 + 81 = 162 81 + -18k + k2 = 162 Factor a perfect square on the left side: (k + -9)(k + -9) = 162 Calculate the square root of the right side: 12.727922061 Break this problem into two subproblems by setting (k + -9) equal to 12.727922061 and -12.727922061.Subproblem 1
k + -9 = 12.727922061 Simplifying k + -9 = 12.727922061 Reorder the terms: -9 + k = 12.727922061 Solving -9 + k = 12.727922061 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + k = 12.727922061 + 9 Combine like terms: -9 + 9 = 0 0 + k = 12.727922061 + 9 k = 12.727922061 + 9 Combine like terms: 12.727922061 + 9 = 21.727922061 k = 21.727922061 Simplifying k = 21.727922061Subproblem 2
k + -9 = -12.727922061 Simplifying k + -9 = -12.727922061 Reorder the terms: -9 + k = -12.727922061 Solving -9 + k = -12.727922061 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + k = -12.727922061 + 9 Combine like terms: -9 + 9 = 0 0 + k = -12.727922061 + 9 k = -12.727922061 + 9 Combine like terms: -12.727922061 + 9 = -3.727922061 k = -3.727922061 Simplifying k = -3.727922061Solution
The solution to the problem is based on the solutions from the subproblems. k = {21.727922061, -3.727922061}Solution
k = {0, 21.727922061, -3.727922061}
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