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Simplifying 8k2 + -20 = -9k Reorder the terms: -20 + 8k2 = -9k Solving -20 + 8k2 = -9k Solving for variable 'k'. Reorder the terms: -20 + 9k + 8k2 = -9k + 9k Combine like terms: -9k + 9k = 0 -20 + 9k + 8k2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.5 + 1.125k + k2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 1.125k + 2.5 + k2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 1.125k + k2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 1.125k + k2 = 0 + 2.5 1.125k + k2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 1.125k + k2 = 2.5 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 = 2.5 + 0.31640625 Reorder the terms: 0.31640625 + 1.125k + k2 = 2.5 + 0.31640625 Combine like terms: 2.5 + 0.31640625 = 2.81640625 0.31640625 + 1.125k + k2 = 2.81640625 Factor a perfect square on the left side: (k + 0.5625)(k + 0.5625) = 2.81640625 Calculate the square root of the right side: 1.678215198 Break this problem into two subproblems by setting (k + 0.5625) equal to 1.678215198 and -1.678215198.Subproblem 1
k + 0.5625 = 1.678215198 Simplifying k + 0.5625 = 1.678215198 Reorder the terms: 0.5625 + k = 1.678215198 Solving 0.5625 + k = 1.678215198 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 = 1.678215198 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + k = 1.678215198 + -0.5625 k = 1.678215198 + -0.5625 Combine like terms: 1.678215198 + -0.5625 = 1.115715198 k = 1.115715198 Simplifying k = 1.115715198Subproblem 2
k + 0.5625 = -1.678215198 Simplifying k + 0.5625 = -1.678215198 Reorder the terms: 0.5625 + k = -1.678215198 Solving 0.5625 + k = -1.678215198 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 = -1.678215198 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + k = -1.678215198 + -0.5625 k = -1.678215198 + -0.5625 Combine like terms: -1.678215198 + -0.5625 = -2.240715198 k = -2.240715198 Simplifying k = -2.240715198Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.115715198, -2.240715198}
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