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Simplifying 8k2 + -9k + -5 = 0 Reorder the terms: -5 + -9k + 8k2 = 0 Solving -5 + -9k + 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'. -0.625 + -1.125k + k2 = 0 Move the constant term to the right: Add '0.625' to each side of the equation. -0.625 + -1.125k + 0.625 + k2 = 0 + 0.625 Reorder the terms: -0.625 + 0.625 + -1.125k + k2 = 0 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + -1.125k + k2 = 0 + 0.625 -1.125k + k2 = 0 + 0.625 Combine like terms: 0 + 0.625 = 0.625 -1.125k + k2 = 0.625 The k term is -1.125k. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125k + 0.31640625 + k2 = 0.625 + 0.31640625 Reorder the terms: 0.31640625 + -1.125k + k2 = 0.625 + 0.31640625 Combine like terms: 0.625 + 0.31640625 = 0.94140625 0.31640625 + -1.125k + k2 = 0.94140625 Factor a perfect square on the left side: (k + -0.5625)(k + -0.5625) = 0.94140625 Calculate the square root of the right side: 0.970260919 Break this problem into two subproblems by setting (k + -0.5625) equal to 0.970260919 and -0.970260919.Subproblem 1
k + -0.5625 = 0.970260919 Simplifying k + -0.5625 = 0.970260919 Reorder the terms: -0.5625 + k = 0.970260919 Solving -0.5625 + k = 0.970260919 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + k = 0.970260919 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + k = 0.970260919 + 0.5625 k = 0.970260919 + 0.5625 Combine like terms: 0.970260919 + 0.5625 = 1.532760919 k = 1.532760919 Simplifying k = 1.532760919Subproblem 2
k + -0.5625 = -0.970260919 Simplifying k + -0.5625 = -0.970260919 Reorder the terms: -0.5625 + k = -0.970260919 Solving -0.5625 + k = -0.970260919 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + k = -0.970260919 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + k = -0.970260919 + 0.5625 k = -0.970260919 + 0.5625 Combine like terms: -0.970260919 + 0.5625 = -0.407760919 k = -0.407760919 Simplifying k = -0.407760919Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.532760919, -0.407760919}
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