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Simplifying 8k2 + 6k + -12 = 3 Reorder the terms: -12 + 6k + 8k2 = 3 Solving -12 + 6k + 8k2 = 3 Solving for variable 'k'. Reorder the terms: -12 + -3 + 6k + 8k2 = 3 + -3 Combine like terms: -12 + -3 = -15 -15 + 6k + 8k2 = 3 + -3 Combine like terms: 3 + -3 = 0 -15 + 6k + 8k2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.875 + 0.75k + k2 = 0 Move the constant term to the right: Add '1.875' to each side of the equation. -1.875 + 0.75k + 1.875 + k2 = 0 + 1.875 Reorder the terms: -1.875 + 1.875 + 0.75k + k2 = 0 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + 0.75k + k2 = 0 + 1.875 0.75k + k2 = 0 + 1.875 Combine like terms: 0 + 1.875 = 1.875 0.75k + k2 = 1.875 The k term is 0.75k. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75k + 0.140625 + k2 = 1.875 + 0.140625 Reorder the terms: 0.140625 + 0.75k + k2 = 1.875 + 0.140625 Combine like terms: 1.875 + 0.140625 = 2.015625 0.140625 + 0.75k + k2 = 2.015625 Factor a perfect square on the left side: (k + 0.375)(k + 0.375) = 2.015625 Calculate the square root of the right side: 1.419727086 Break this problem into two subproblems by setting (k + 0.375) equal to 1.419727086 and -1.419727086.Subproblem 1
k + 0.375 = 1.419727086 Simplifying k + 0.375 = 1.419727086 Reorder the terms: 0.375 + k = 1.419727086 Solving 0.375 + k = 1.419727086 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + k = 1.419727086 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + k = 1.419727086 + -0.375 k = 1.419727086 + -0.375 Combine like terms: 1.419727086 + -0.375 = 1.044727086 k = 1.044727086 Simplifying k = 1.044727086Subproblem 2
k + 0.375 = -1.419727086 Simplifying k + 0.375 = -1.419727086 Reorder the terms: 0.375 + k = -1.419727086 Solving 0.375 + k = -1.419727086 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + k = -1.419727086 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + k = -1.419727086 + -0.375 k = -1.419727086 + -0.375 Combine like terms: -1.419727086 + -0.375 = -1.794727086 k = -1.794727086 Simplifying k = -1.794727086Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.044727086, -1.794727086}
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