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Simplifying 9k2 + 6k + -9 = 0 Reorder the terms: -9 + 6k + 9k2 = 0 Solving -9 + 6k + 9k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '3'. 3(-3 + 2k + 3k2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-3 + 2k + 3k2)' equal to zero and attempt to solve: Simplifying -3 + 2k + 3k2 = 0 Solving -3 + 2k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1 + 0.6666666667k + k2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.6666666667k + 1 + k2 = 0 + 1 Reorder the terms: -1 + 1 + 0.6666666667k + k2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.6666666667k + k2 = 0 + 1 0.6666666667k + k2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.6666666667k + k2 = 1 The k term is 0.6666666667k. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667k + 0.1111111112 + k2 = 1 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667k + k2 = 1 + 0.1111111112 Combine like terms: 1 + 0.1111111112 = 1.1111111112 0.1111111112 + 0.6666666667k + k2 = 1.1111111112 Factor a perfect square on the left side: (k + 0.3333333334)(k + 0.3333333334) = 1.1111111112 Calculate the square root of the right side: 1.054092553 Break this problem into two subproblems by setting (k + 0.3333333334) equal to 1.054092553 and -1.054092553.Subproblem 1
k + 0.3333333334 = 1.054092553 Simplifying k + 0.3333333334 = 1.054092553 Reorder the terms: 0.3333333334 + k = 1.054092553 Solving 0.3333333334 + k = 1.054092553 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + k = 1.054092553 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + k = 1.054092553 + -0.3333333334 k = 1.054092553 + -0.3333333334 Combine like terms: 1.054092553 + -0.3333333334 = 0.7207592196 k = 0.7207592196 Simplifying k = 0.7207592196Subproblem 2
k + 0.3333333334 = -1.054092553 Simplifying k + 0.3333333334 = -1.054092553 Reorder the terms: 0.3333333334 + k = -1.054092553 Solving 0.3333333334 + k = -1.054092553 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + k = -1.054092553 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + k = -1.054092553 + -0.3333333334 k = -1.054092553 + -0.3333333334 Combine like terms: -1.054092553 + -0.3333333334 = -1.3874258864 k = -1.3874258864 Simplifying k = -1.3874258864Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.7207592196, -1.3874258864}Solution
k = {0.7207592196, -1.3874258864}
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