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Simplifying 8k2 + -12k + 1 = 0 Reorder the terms: 1 + -12k + 8k2 = 0 Solving 1 + -12k + 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'. 0.125 + -1.5k + k2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + -1.5k + -0.125 + k2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + -1.5k + k2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + -1.5k + k2 = 0 + -0.125 -1.5k + k2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 -1.5k + k2 = -0.125 The k term is -1.5k. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5k + 0.5625 + k2 = -0.125 + 0.5625 Reorder the terms: 0.5625 + -1.5k + k2 = -0.125 + 0.5625 Combine like terms: -0.125 + 0.5625 = 0.4375 0.5625 + -1.5k + k2 = 0.4375 Factor a perfect square on the left side: (k + -0.75)(k + -0.75) = 0.4375 Calculate the square root of the right side: 0.661437828 Break this problem into two subproblems by setting (k + -0.75) equal to 0.661437828 and -0.661437828.Subproblem 1
k + -0.75 = 0.661437828 Simplifying k + -0.75 = 0.661437828 Reorder the terms: -0.75 + k = 0.661437828 Solving -0.75 + k = 0.661437828 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + k = 0.661437828 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + k = 0.661437828 + 0.75 k = 0.661437828 + 0.75 Combine like terms: 0.661437828 + 0.75 = 1.411437828 k = 1.411437828 Simplifying k = 1.411437828Subproblem 2
k + -0.75 = -0.661437828 Simplifying k + -0.75 = -0.661437828 Reorder the terms: -0.75 + k = -0.661437828 Solving -0.75 + k = -0.661437828 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + k = -0.661437828 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + k = -0.661437828 + 0.75 k = -0.661437828 + 0.75 Combine like terms: -0.661437828 + 0.75 = 0.088562172 k = 0.088562172 Simplifying k = 0.088562172Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.411437828, 0.088562172}
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