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Simplifying 20k2 + 12k3 + -20k = 0 Reorder the terms: -20k + 20k2 + 12k3 = 0 Solving -20k + 20k2 + 12k3 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '4k'. 4k(-5 + 5k + 3k2) = 0 Ignore the factor 4.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 '(-5 + 5k + 3k2)' equal to zero and attempt to solve: Simplifying -5 + 5k + 3k2 = 0 Solving -5 + 5k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.666666667 + 1.666666667k + k2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667k + 1.666666667 + k2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 1.666666667k + k2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 1.666666667k + k2 = 0 + 1.666666667 1.666666667k + k2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 1.666666667k + k2 = 1.666666667 The k term is 1.666666667k. Take half its coefficient (0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. 1.666666667k + 0.6944444447 + k2 = 1.666666667 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667k + k2 = 1.666666667 + 0.6944444447 Combine like terms: 1.666666667 + 0.6944444447 = 2.3611111117 0.6944444447 + 1.666666667k + k2 = 2.3611111117 Factor a perfect square on the left side: (k + 0.8333333335)(k + 0.8333333335) = 2.3611111117 Calculate the square root of the right side: 1.536590743 Break this problem into two subproblems by setting (k + 0.8333333335) equal to 1.536590743 and -1.536590743.Subproblem 1
k + 0.8333333335 = 1.536590743 Simplifying k + 0.8333333335 = 1.536590743 Reorder the terms: 0.8333333335 + k = 1.536590743 Solving 0.8333333335 + k = 1.536590743 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + k = 1.536590743 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + k = 1.536590743 + -0.8333333335 k = 1.536590743 + -0.8333333335 Combine like terms: 1.536590743 + -0.8333333335 = 0.7032574095 k = 0.7032574095 Simplifying k = 0.7032574095Subproblem 2
k + 0.8333333335 = -1.536590743 Simplifying k + 0.8333333335 = -1.536590743 Reorder the terms: 0.8333333335 + k = -1.536590743 Solving 0.8333333335 + k = -1.536590743 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + k = -1.536590743 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + k = -1.536590743 + -0.8333333335 k = -1.536590743 + -0.8333333335 Combine like terms: -1.536590743 + -0.8333333335 = -2.3699240765 k = -2.3699240765 Simplifying k = -2.3699240765Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.7032574095, -2.3699240765}Solution
k = {0, 0.7032574095, -2.3699240765}
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