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Simplifying 11k2 = 1 + -6k Solving 11k2 = 1 + -6k Solving for variable 'k'. Reorder the terms: -1 + 6k + 11k2 = 1 + -6k + -1 + 6k Reorder the terms: -1 + 6k + 11k2 = 1 + -1 + -6k + 6k Combine like terms: 1 + -1 = 0 -1 + 6k + 11k2 = 0 + -6k + 6k -1 + 6k + 11k2 = -6k + 6k Combine like terms: -6k + 6k = 0 -1 + 6k + 11k2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -0.09090909091 + 0.5454545455k + k2 = 0 Move the constant term to the right: Add '0.09090909091' to each side of the equation. -0.09090909091 + 0.5454545455k + 0.09090909091 + k2 = 0 + 0.09090909091 Reorder the terms: -0.09090909091 + 0.09090909091 + 0.5454545455k + k2 = 0 + 0.09090909091 Combine like terms: -0.09090909091 + 0.09090909091 = 0.00000000000 0.00000000000 + 0.5454545455k + k2 = 0 + 0.09090909091 0.5454545455k + k2 = 0 + 0.09090909091 Combine like terms: 0 + 0.09090909091 = 0.09090909091 0.5454545455k + k2 = 0.09090909091 The k term is 0.5454545455k. Take half its coefficient (0.2727272728). Square it (0.07438016533) and add it to both sides. Add '0.07438016533' to each side of the equation. 0.5454545455k + 0.07438016533 + k2 = 0.09090909091 + 0.07438016533 Reorder the terms: 0.07438016533 + 0.5454545455k + k2 = 0.09090909091 + 0.07438016533 Combine like terms: 0.09090909091 + 0.07438016533 = 0.16528925624 0.07438016533 + 0.5454545455k + k2 = 0.16528925624 Factor a perfect square on the left side: (k + 0.2727272728)(k + 0.2727272728) = 0.16528925624 Calculate the square root of the right side: 0.406557814 Break this problem into two subproblems by setting (k + 0.2727272728) equal to 0.406557814 and -0.406557814.Subproblem 1
k + 0.2727272728 = 0.406557814 Simplifying k + 0.2727272728 = 0.406557814 Reorder the terms: 0.2727272728 + k = 0.406557814 Solving 0.2727272728 + k = 0.406557814 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.2727272728' to each side of the equation. 0.2727272728 + -0.2727272728 + k = 0.406557814 + -0.2727272728 Combine like terms: 0.2727272728 + -0.2727272728 = 0.0000000000 0.0000000000 + k = 0.406557814 + -0.2727272728 k = 0.406557814 + -0.2727272728 Combine like terms: 0.406557814 + -0.2727272728 = 0.1338305412 k = 0.1338305412 Simplifying k = 0.1338305412Subproblem 2
k + 0.2727272728 = -0.406557814 Simplifying k + 0.2727272728 = -0.406557814 Reorder the terms: 0.2727272728 + k = -0.406557814 Solving 0.2727272728 + k = -0.406557814 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.2727272728' to each side of the equation. 0.2727272728 + -0.2727272728 + k = -0.406557814 + -0.2727272728 Combine like terms: 0.2727272728 + -0.2727272728 = 0.0000000000 0.0000000000 + k = -0.406557814 + -0.2727272728 k = -0.406557814 + -0.2727272728 Combine like terms: -0.406557814 + -0.2727272728 = -0.6792850868 k = -0.6792850868 Simplifying k = -0.6792850868Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.1338305412, -0.6792850868}
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