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Simplifying 9k2 + 9k + -3 = 0 Reorder the terms: -3 + 9k + 9k2 = 0 Solving -3 + 9k + 9k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '3'. 3(-1 + 3k + 3k2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-1 + 3k + 3k2)' equal to zero and attempt to solve: Simplifying -1 + 3k + 3k2 = 0 Solving -1 + 3k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + k + k2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + k + 0.3333333333 + k2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + k + k2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + k + k2 = 0 + 0.3333333333 k + k2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 k + k2 = 0.3333333333 The k term is k. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + k2 = 0.3333333333 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + k2 = 0.3333333333 + 0.25 Combine like terms: 0.3333333333 + 0.25 = 0.5833333333 1.25 + k2 = 0.5833333333 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 0.5833333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (k + 0.5) equal to 0.763762616 and -0.763762616.Subproblem 1
k + 0.5 = 0.763762616 Simplifying k + 0.5 = 0.763762616 Reorder the terms: 0.5 + k = 0.763762616 Solving 0.5 + k = 0.763762616 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = 0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 0.763762616 + -0.5 k = 0.763762616 + -0.5 Combine like terms: 0.763762616 + -0.5 = 0.263762616 k = 0.263762616 Simplifying k = 0.263762616Subproblem 2
k + 0.5 = -0.763762616 Simplifying k + 0.5 = -0.763762616 Reorder the terms: 0.5 + k = -0.763762616 Solving 0.5 + k = -0.763762616 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = -0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -0.763762616 + -0.5 k = -0.763762616 + -0.5 Combine like terms: -0.763762616 + -0.5 = -1.263762616 k = -1.263762616 Simplifying k = -1.263762616Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.263762616, -1.263762616}Solution
k = {0.263762616, -1.263762616}
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