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Simplifying 5k2(-6k2 + -2k + 6) = 0 Reorder the terms: 5k2(6 + -2k + -6k2) = 0 (6 * 5k2 + -2k * 5k2 + -6k2 * 5k2) = 0 (30k2 + -10k3 + -30k4) = 0 Solving 30k2 + -10k3 + -30k4 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '10k2'. 10k2(3 + -1k + -3k2) = 0 Ignore the factor 10.Subproblem 1
Set the factor 'k2' equal to zero and attempt to solve: Simplifying k2 = 0 Solving k2 = 0 Move all terms containing k to the left, all other terms to the right. Simplifying k2 = 0 Take the square root of each side: k = {0}Subproblem 2
Set the factor '(3 + -1k + -3k2)' equal to zero and attempt to solve: Simplifying 3 + -1k + -3k2 = 0 Solving 3 + -1k + -3k2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -1 + 0.3333333333k + k2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.3333333333k + 1 + k2 = 0 + 1 Reorder the terms: -1 + 1 + 0.3333333333k + k2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.3333333333k + k2 = 0 + 1 0.3333333333k + k2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.3333333333k + k2 = 1 The k term is 0.3333333333k. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333k + 0.02777777779 + k2 = 1 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333k + k2 = 1 + 0.02777777779 Combine like terms: 1 + 0.02777777779 = 1.02777777779 0.02777777779 + 0.3333333333k + k2 = 1.02777777779 Factor a perfect square on the left side: (k + 0.1666666667)(k + 0.1666666667) = 1.02777777779 Calculate the square root of the right side: 1.013793755 Break this problem into two subproblems by setting (k + 0.1666666667) equal to 1.013793755 and -1.013793755.Subproblem 1
k + 0.1666666667 = 1.013793755 Simplifying k + 0.1666666667 = 1.013793755 Reorder the terms: 0.1666666667 + k = 1.013793755 Solving 0.1666666667 + k = 1.013793755 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + k = 1.013793755 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + k = 1.013793755 + -0.1666666667 k = 1.013793755 + -0.1666666667 Combine like terms: 1.013793755 + -0.1666666667 = 0.8471270883 k = 0.8471270883 Simplifying k = 0.8471270883Subproblem 2
k + 0.1666666667 = -1.013793755 Simplifying k + 0.1666666667 = -1.013793755 Reorder the terms: 0.1666666667 + k = -1.013793755 Solving 0.1666666667 + k = -1.013793755 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + k = -1.013793755 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + k = -1.013793755 + -0.1666666667 k = -1.013793755 + -0.1666666667 Combine like terms: -1.013793755 + -0.1666666667 = -1.1804604217 k = -1.1804604217 Simplifying k = -1.1804604217Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.8471270883, -1.1804604217}Solution
k = {0, 0.8471270883, -1.1804604217}
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