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Simplifying 2k2 + 1k + -2 = 0 Reorder the terms: -2 + 1k + 2k2 = 0 Solving -2 + 1k + 2k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1 + 0.5k + k2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.5k + 1 + k2 = 0 + 1 Reorder the terms: -1 + 1 + 0.5k + k2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.5k + k2 = 0 + 1 0.5k + k2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.5k + k2 = 1 The k term is 0.5k. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5k + 0.0625 + k2 = 1 + 0.0625 Reorder the terms: 0.0625 + 0.5k + k2 = 1 + 0.0625 Combine like terms: 1 + 0.0625 = 1.0625 0.0625 + 0.5k + k2 = 1.0625 Factor a perfect square on the left side: (k + 0.25)(k + 0.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (k + 0.25) equal to 1.030776406 and -1.030776406.Subproblem 1
k + 0.25 = 1.030776406 Simplifying k + 0.25 = 1.030776406 Reorder the terms: 0.25 + k = 1.030776406 Solving 0.25 + k = 1.030776406 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + k = 1.030776406 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + k = 1.030776406 + -0.25 k = 1.030776406 + -0.25 Combine like terms: 1.030776406 + -0.25 = 0.780776406 k = 0.780776406 Simplifying k = 0.780776406Subproblem 2
k + 0.25 = -1.030776406 Simplifying k + 0.25 = -1.030776406 Reorder the terms: 0.25 + k = -1.030776406 Solving 0.25 + k = -1.030776406 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + k = -1.030776406 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + k = -1.030776406 + -0.25 k = -1.030776406 + -0.25 Combine like terms: -1.030776406 + -0.25 = -1.280776406 k = -1.280776406 Simplifying k = -1.280776406Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.780776406, -1.280776406}
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