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Simplifying 6k2 = 19 + -6k Solving 6k2 = 19 + -6k Solving for variable 'k'. Reorder the terms: -19 + 6k + 6k2 = 19 + -6k + -19 + 6k Reorder the terms: -19 + 6k + 6k2 = 19 + -19 + -6k + 6k Combine like terms: 19 + -19 = 0 -19 + 6k + 6k2 = 0 + -6k + 6k -19 + 6k + 6k2 = -6k + 6k Combine like terms: -6k + 6k = 0 -19 + 6k + 6k2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -3.166666667 + k + k2 = 0 Move the constant term to the right: Add '3.166666667' to each side of the equation. -3.166666667 + k + 3.166666667 + k2 = 0 + 3.166666667 Reorder the terms: -3.166666667 + 3.166666667 + k + k2 = 0 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + k + k2 = 0 + 3.166666667 k + k2 = 0 + 3.166666667 Combine like terms: 0 + 3.166666667 = 3.166666667 k + k2 = 3.166666667 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 = 3.166666667 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + k2 = 3.166666667 + 0.25 Combine like terms: 3.166666667 + 0.25 = 3.416666667 1.25 + k2 = 3.416666667 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 3.416666667 Calculate the square root of the right side: 1.848422751 Break this problem into two subproblems by setting (k + 0.5) equal to 1.848422751 and -1.848422751.Subproblem 1
k + 0.5 = 1.848422751 Simplifying k + 0.5 = 1.848422751 Reorder the terms: 0.5 + k = 1.848422751 Solving 0.5 + k = 1.848422751 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 = 1.848422751 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 1.848422751 + -0.5 k = 1.848422751 + -0.5 Combine like terms: 1.848422751 + -0.5 = 1.348422751 k = 1.348422751 Simplifying k = 1.348422751Subproblem 2
k + 0.5 = -1.848422751 Simplifying k + 0.5 = -1.848422751 Reorder the terms: 0.5 + k = -1.848422751 Solving 0.5 + k = -1.848422751 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 = -1.848422751 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -1.848422751 + -0.5 k = -1.848422751 + -0.5 Combine like terms: -1.848422751 + -0.5 = -2.348422751 k = -2.348422751 Simplifying k = -2.348422751Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.348422751, -2.348422751}
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