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Simplifying 3k2 + 9k + 5 = 0 Reorder the terms: 5 + 9k + 3k2 = 0 Solving 5 + 9k + 3k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.666666667 + 3k + k2 = 0 Move the constant term to the right: Add '-1.666666667' to each side of the equation. 1.666666667 + 3k + -1.666666667 + k2 = 0 + -1.666666667 Reorder the terms: 1.666666667 + -1.666666667 + 3k + k2 = 0 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + 3k + k2 = 0 + -1.666666667 3k + k2 = 0 + -1.666666667 Combine like terms: 0 + -1.666666667 = -1.666666667 3k + k2 = -1.666666667 The k term is 3k. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3k + 2.25 + k2 = -1.666666667 + 2.25 Reorder the terms: 2.25 + 3k + k2 = -1.666666667 + 2.25 Combine like terms: -1.666666667 + 2.25 = 0.583333333 2.25 + 3k + k2 = 0.583333333 Factor a perfect square on the left side: (k + 1.5)(k + 1.5) = 0.583333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (k + 1.5) equal to 0.763762616 and -0.763762616.Subproblem 1
k + 1.5 = 0.763762616 Simplifying k + 1.5 = 0.763762616 Reorder the terms: 1.5 + k = 0.763762616 Solving 1.5 + k = 0.763762616 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + k = 0.763762616 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + k = 0.763762616 + -1.5 k = 0.763762616 + -1.5 Combine like terms: 0.763762616 + -1.5 = -0.736237384 k = -0.736237384 Simplifying k = -0.736237384Subproblem 2
k + 1.5 = -0.763762616 Simplifying k + 1.5 = -0.763762616 Reorder the terms: 1.5 + k = -0.763762616 Solving 1.5 + k = -0.763762616 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + k = -0.763762616 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + k = -0.763762616 + -1.5 k = -0.763762616 + -1.5 Combine like terms: -0.763762616 + -1.5 = -2.263762616 k = -2.263762616 Simplifying k = -2.263762616Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.736237384, -2.263762616}
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