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Simplifying -6k2 + 7k + 8 = 0 Reorder the terms: 8 + 7k + -6k2 = 0 Solving 8 + 7k + -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'. -1.333333333 + -1.166666667k + k2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + -1.166666667k + 1.333333333 + k2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + -1.166666667k + k2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + -1.166666667k + k2 = 0 + 1.333333333 -1.166666667k + k2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 -1.166666667k + k2 = 1.333333333 The k term is -1.166666667k. Take half its coefficient (-0.5833333335). Square it (0.3402777780) and add it to both sides. Add '0.3402777780' to each side of the equation. -1.166666667k + 0.3402777780 + k2 = 1.333333333 + 0.3402777780 Reorder the terms: 0.3402777780 + -1.166666667k + k2 = 1.333333333 + 0.3402777780 Combine like terms: 1.333333333 + 0.3402777780 = 1.673611111 0.3402777780 + -1.166666667k + k2 = 1.673611111 Factor a perfect square on the left side: (k + -0.5833333335)(k + -0.5833333335) = 1.673611111 Calculate the square root of the right side: 1.293681225 Break this problem into two subproblems by setting (k + -0.5833333335) equal to 1.293681225 and -1.293681225.Subproblem 1
k + -0.5833333335 = 1.293681225 Simplifying k + -0.5833333335 = 1.293681225 Reorder the terms: -0.5833333335 + k = 1.293681225 Solving -0.5833333335 + k = 1.293681225 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5833333335' to each side of the equation. -0.5833333335 + 0.5833333335 + k = 1.293681225 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + k = 1.293681225 + 0.5833333335 k = 1.293681225 + 0.5833333335 Combine like terms: 1.293681225 + 0.5833333335 = 1.8770145585 k = 1.8770145585 Simplifying k = 1.8770145585Subproblem 2
k + -0.5833333335 = -1.293681225 Simplifying k + -0.5833333335 = -1.293681225 Reorder the terms: -0.5833333335 + k = -1.293681225 Solving -0.5833333335 + k = -1.293681225 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5833333335' to each side of the equation. -0.5833333335 + 0.5833333335 + k = -1.293681225 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + k = -1.293681225 + 0.5833333335 k = -1.293681225 + 0.5833333335 Combine like terms: -1.293681225 + 0.5833333335 = -0.7103478915 k = -0.7103478915 Simplifying k = -0.7103478915Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.8770145585, -0.7103478915}
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