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Simplifying 4k2 + 2k = 1 Reorder the terms: 2k + 4k2 = 1 Solving 2k + 4k2 = 1 Solving for variable 'k'. Reorder the terms: -1 + 2k + 4k2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 2k + 4k2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + 0.5k + k2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 0.5k + 0.25 + k2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 0.5k + k2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 0.5k + k2 = 0 + 0.25 0.5k + k2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 0.5k + k2 = 0.25 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 = 0.25 + 0.0625 Reorder the terms: 0.0625 + 0.5k + k2 = 0.25 + 0.0625 Combine like terms: 0.25 + 0.0625 = 0.3125 0.0625 + 0.5k + k2 = 0.3125 Factor a perfect square on the left side: (k + 0.25)(k + 0.25) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (k + 0.25) equal to 0.559016994 and -0.559016994.Subproblem 1
k + 0.25 = 0.559016994 Simplifying k + 0.25 = 0.559016994 Reorder the terms: 0.25 + k = 0.559016994 Solving 0.25 + k = 0.559016994 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 = 0.559016994 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + k = 0.559016994 + -0.25 k = 0.559016994 + -0.25 Combine like terms: 0.559016994 + -0.25 = 0.309016994 k = 0.309016994 Simplifying k = 0.309016994Subproblem 2
k + 0.25 = -0.559016994 Simplifying k + 0.25 = -0.559016994 Reorder the terms: 0.25 + k = -0.559016994 Solving 0.25 + k = -0.559016994 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 = -0.559016994 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + k = -0.559016994 + -0.25 k = -0.559016994 + -0.25 Combine like terms: -0.559016994 + -0.25 = -0.809016994 k = -0.809016994 Simplifying k = -0.809016994Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.309016994, -0.809016994}
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