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Simplifying 8k2 + -19k + 9 = 0 Reorder the terms: 9 + -19k + 8k2 = 0 Solving 9 + -19k + 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 + -2.375k + k2 = 0 Move the constant term to the right: Add '-1.125' to each side of the equation. 1.125 + -2.375k + -1.125 + k2 = 0 + -1.125 Reorder the terms: 1.125 + -1.125 + -2.375k + k2 = 0 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + -2.375k + k2 = 0 + -1.125 -2.375k + k2 = 0 + -1.125 Combine like terms: 0 + -1.125 = -1.125 -2.375k + k2 = -1.125 The k term is -2.375k. Take half its coefficient (-1.1875). Square it (1.41015625) and add it to both sides. Add '1.41015625' to each side of the equation. -2.375k + 1.41015625 + k2 = -1.125 + 1.41015625 Reorder the terms: 1.41015625 + -2.375k + k2 = -1.125 + 1.41015625 Combine like terms: -1.125 + 1.41015625 = 0.28515625 1.41015625 + -2.375k + k2 = 0.28515625 Factor a perfect square on the left side: (k + -1.1875)(k + -1.1875) = 0.28515625 Calculate the square root of the right side: 0.534000234 Break this problem into two subproblems by setting (k + -1.1875) equal to 0.534000234 and -0.534000234.Subproblem 1
k + -1.1875 = 0.534000234 Simplifying k + -1.1875 = 0.534000234 Reorder the terms: -1.1875 + k = 0.534000234 Solving -1.1875 + k = 0.534000234 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + k = 0.534000234 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + k = 0.534000234 + 1.1875 k = 0.534000234 + 1.1875 Combine like terms: 0.534000234 + 1.1875 = 1.721500234 k = 1.721500234 Simplifying k = 1.721500234Subproblem 2
k + -1.1875 = -0.534000234 Simplifying k + -1.1875 = -0.534000234 Reorder the terms: -1.1875 + k = -0.534000234 Solving -1.1875 + k = -0.534000234 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + k = -0.534000234 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + k = -0.534000234 + 1.1875 k = -0.534000234 + 1.1875 Combine like terms: -0.534000234 + 1.1875 = 0.653499766 k = 0.653499766 Simplifying k = 0.653499766Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.721500234, 0.653499766}
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