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Simplifying 3k2 + -6 = k Reorder the terms: -6 + 3k2 = k Solving -6 + 3k2 = k Solving for variable 'k'. Reorder the terms: -6 + -1k + 3k2 = k + -1k Combine like terms: k + -1k = 0 -6 + -1k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2 + -0.3333333333k + k2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -0.3333333333k + 2 + k2 = 0 + 2 Reorder the terms: -2 + 2 + -0.3333333333k + k2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -0.3333333333k + k2 = 0 + 2 -0.3333333333k + k2 = 0 + 2 Combine like terms: 0 + 2 = 2 -0.3333333333k + k2 = 2 The k term is -0.3333333333k. Take half its coefficient (-0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. -0.3333333333k + 0.02777777779 + k2 = 2 + 0.02777777779 Reorder the terms: 0.02777777779 + -0.3333333333k + k2 = 2 + 0.02777777779 Combine like terms: 2 + 0.02777777779 = 2.02777777779 0.02777777779 + -0.3333333333k + k2 = 2.02777777779 Factor a perfect square on the left side: (k + -0.1666666667)(k + -0.1666666667) = 2.02777777779 Calculate the square root of the right side: 1.424000624 Break this problem into two subproblems by setting (k + -0.1666666667) equal to 1.424000624 and -1.424000624.Subproblem 1
k + -0.1666666667 = 1.424000624 Simplifying k + -0.1666666667 = 1.424000624 Reorder the terms: -0.1666666667 + k = 1.424000624 Solving -0.1666666667 + k = 1.424000624 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + k = 1.424000624 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + k = 1.424000624 + 0.1666666667 k = 1.424000624 + 0.1666666667 Combine like terms: 1.424000624 + 0.1666666667 = 1.5906672907 k = 1.5906672907 Simplifying k = 1.5906672907Subproblem 2
k + -0.1666666667 = -1.424000624 Simplifying k + -0.1666666667 = -1.424000624 Reorder the terms: -0.1666666667 + k = -1.424000624 Solving -0.1666666667 + k = -1.424000624 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + k = -1.424000624 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + k = -1.424000624 + 0.1666666667 k = -1.424000624 + 0.1666666667 Combine like terms: -1.424000624 + 0.1666666667 = -1.2573339573 k = -1.2573339573 Simplifying k = -1.2573339573Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.5906672907, -1.2573339573}
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