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Simplifying 3k2 + -9k = -5 Reorder the terms: -9k + 3k2 = -5 Solving -9k + 3k2 = -5 Solving for variable 'k'. Reorder the terms: 5 + -9k + 3k2 = -5 + 5 Combine like terms: -5 + 5 = 0 5 + -9k + 3k2 = 0 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 = 2.263762616 k = 2.263762616 Simplifying k = 2.263762616Subproblem 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 = 0.736237384 k = 0.736237384 Simplifying k = 0.736237384Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.263762616, 0.736237384}
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