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Simplifying 6k2 + 6k + -5 = 0 Reorder the terms: -5 + 6k + 6k2 = 0 Solving -5 + 6k + 6k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.8333333333 + k + k2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + k + 0.8333333333 + k2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + k + k2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + k + k2 = 0 + 0.8333333333 k + k2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 k + k2 = 0.8333333333 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 = 0.8333333333 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + k2 = 0.8333333333 + 0.25 Combine like terms: 0.8333333333 + 0.25 = 1.0833333333 1.25 + k2 = 1.0833333333 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 1.0833333333 Calculate the square root of the right side: 1.040833 Break this problem into two subproblems by setting (k + 0.5) equal to 1.040833 and -1.040833.Subproblem 1
k + 0.5 = 1.040833 Simplifying k + 0.5 = 1.040833 Reorder the terms: 0.5 + k = 1.040833 Solving 0.5 + k = 1.040833 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 = 1.040833 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 1.040833 + -0.5 k = 1.040833 + -0.5 Combine like terms: 1.040833 + -0.5 = 0.540833 k = 0.540833 Simplifying k = 0.540833Subproblem 2
k + 0.5 = -1.040833 Simplifying k + 0.5 = -1.040833 Reorder the terms: 0.5 + k = -1.040833 Solving 0.5 + k = -1.040833 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 = -1.040833 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -1.040833 + -0.5 k = -1.040833 + -0.5 Combine like terms: -1.040833 + -0.5 = -1.540833 k = -1.540833 Simplifying k = -1.540833Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.540833, -1.540833}
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