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Simplifying 6k2 + -6k + -1 = 0 Reorder the terms: -1 + -6k + 6k2 = 0 Solving -1 + -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'. -0.1666666667 + -1k + k2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + -1k + 0.1666666667 + k2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + -1k + k2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + -1k + k2 = 0 + 0.1666666667 -1k + k2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 -1k + k2 = 0.1666666667 The k term is -1k. 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. -1k + 0.25 + k2 = 0.1666666667 + 0.25 Reorder the terms: 0.25 + -1k + k2 = 0.1666666667 + 0.25 Combine like terms: 0.1666666667 + 0.25 = 0.4166666667 0.25 + -1k + k2 = 0.4166666667 Factor a perfect square on the left side: (k + -0.5)(k + -0.5) = 0.4166666667 Calculate the square root of the right side: 0.645497224 Break this problem into two subproblems by setting (k + -0.5) equal to 0.645497224 and -0.645497224.Subproblem 1
k + -0.5 = 0.645497224 Simplifying k + -0.5 = 0.645497224 Reorder the terms: -0.5 + k = 0.645497224 Solving -0.5 + k = 0.645497224 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 = 0.645497224 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + k = 0.645497224 + 0.5 k = 0.645497224 + 0.5 Combine like terms: 0.645497224 + 0.5 = 1.145497224 k = 1.145497224 Simplifying k = 1.145497224Subproblem 2
k + -0.5 = -0.645497224 Simplifying k + -0.5 = -0.645497224 Reorder the terms: -0.5 + k = -0.645497224 Solving -0.5 + k = -0.645497224 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 = -0.645497224 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + k = -0.645497224 + 0.5 k = -0.645497224 + 0.5 Combine like terms: -0.645497224 + 0.5 = -0.145497224 k = -0.145497224 Simplifying k = -0.145497224Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.145497224, -0.145497224}
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