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Simplifying 8k2 + 12k + -14 = 0 Reorder the terms: -14 + 12k + 8k2 = 0 Solving -14 + 12k + 8k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + 6k + 4k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + 6k + 4k2)' equal to zero and attempt to solve: Simplifying -7 + 6k + 4k2 = 0 Solving -7 + 6k + 4k2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.75 + 1.5k + k2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + 1.5k + 1.75 + k2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + 1.5k + k2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + 1.5k + k2 = 0 + 1.75 1.5k + k2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 1.5k + k2 = 1.75 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 = 1.75 + 0.5625 Reorder the terms: 0.5625 + 1.5k + k2 = 1.75 + 0.5625 Combine like terms: 1.75 + 0.5625 = 2.3125 0.5625 + 1.5k + k2 = 2.3125 Factor a perfect square on the left side: (k + 0.75)(k + 0.75) = 2.3125 Calculate the square root of the right side: 1.520690633 Break this problem into two subproblems by setting (k + 0.75) equal to 1.520690633 and -1.520690633.Subproblem 1
k + 0.75 = 1.520690633 Simplifying k + 0.75 = 1.520690633 Reorder the terms: 0.75 + k = 1.520690633 Solving 0.75 + k = 1.520690633 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 = 1.520690633 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + k = 1.520690633 + -0.75 k = 1.520690633 + -0.75 Combine like terms: 1.520690633 + -0.75 = 0.770690633 k = 0.770690633 Simplifying k = 0.770690633Subproblem 2
k + 0.75 = -1.520690633 Simplifying k + 0.75 = -1.520690633 Reorder the terms: 0.75 + k = -1.520690633 Solving 0.75 + k = -1.520690633 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 = -1.520690633 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + k = -1.520690633 + -0.75 k = -1.520690633 + -0.75 Combine like terms: -1.520690633 + -0.75 = -2.270690633 k = -2.270690633 Simplifying k = -2.270690633Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.770690633, -2.270690633}Solution
k = {0.770690633, -2.270690633}
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