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Simplifying -8k2 + -4k + 9 = 0 Reorder the terms: 9 + -4k + -8k2 = 0 Solving 9 + -4k + -8k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1.125 + 0.5k + k2 = 0 Move the constant term to the right: Add '1.125' to each side of the equation. -1.125 + 0.5k + 1.125 + k2 = 0 + 1.125 Reorder the terms: -1.125 + 1.125 + 0.5k + k2 = 0 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + 0.5k + k2 = 0 + 1.125 0.5k + k2 = 0 + 1.125 Combine like terms: 0 + 1.125 = 1.125 0.5k + k2 = 1.125 The k term is 0.5k. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5k + 0.0625 + k2 = 1.125 + 0.0625 Reorder the terms: 0.0625 + 0.5k + k2 = 1.125 + 0.0625 Combine like terms: 1.125 + 0.0625 = 1.1875 0.0625 + 0.5k + k2 = 1.1875 Factor a perfect square on the left side: (k + 0.25)(k + 0.25) = 1.1875 Calculate the square root of the right side: 1.089724736 Break this problem into two subproblems by setting (k + 0.25) equal to 1.089724736 and -1.089724736.Subproblem 1
k + 0.25 = 1.089724736 Simplifying k + 0.25 = 1.089724736 Reorder the terms: 0.25 + k = 1.089724736 Solving 0.25 + k = 1.089724736 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + k = 1.089724736 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + k = 1.089724736 + -0.25 k = 1.089724736 + -0.25 Combine like terms: 1.089724736 + -0.25 = 0.839724736 k = 0.839724736 Simplifying k = 0.839724736Subproblem 2
k + 0.25 = -1.089724736 Simplifying k + 0.25 = -1.089724736 Reorder the terms: 0.25 + k = -1.089724736 Solving 0.25 + k = -1.089724736 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + k = -1.089724736 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + k = -1.089724736 + -0.25 k = -1.089724736 + -0.25 Combine like terms: -1.089724736 + -0.25 = -1.339724736 k = -1.339724736 Simplifying k = -1.339724736Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.839724736, -1.339724736}
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