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Simplifying k2 + 2k = 11 Reorder the terms: 2k + k2 = 11 Solving 2k + k2 = 11 Solving for variable 'k'. Reorder the terms: -11 + 2k + k2 = 11 + -11 Combine like terms: 11 + -11 = 0 -11 + 2k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '11' to each side of the equation. -11 + 2k + 11 + k2 = 0 + 11 Reorder the terms: -11 + 11 + 2k + k2 = 0 + 11 Combine like terms: -11 + 11 = 0 0 + 2k + k2 = 0 + 11 2k + k2 = 0 + 11 Combine like terms: 0 + 11 = 11 2k + k2 = 11 The k term is 2k. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2k + 1 + k2 = 11 + 1 Reorder the terms: 1 + 2k + k2 = 11 + 1 Combine like terms: 11 + 1 = 12 1 + 2k + k2 = 12 Factor a perfect square on the left side: (k + 1)(k + 1) = 12 Calculate the square root of the right side: 3.464101615 Break this problem into two subproblems by setting (k + 1) equal to 3.464101615 and -3.464101615.Subproblem 1
k + 1 = 3.464101615 Simplifying k + 1 = 3.464101615 Reorder the terms: 1 + k = 3.464101615 Solving 1 + k = 3.464101615 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = 3.464101615 + -1 Combine like terms: 1 + -1 = 0 0 + k = 3.464101615 + -1 k = 3.464101615 + -1 Combine like terms: 3.464101615 + -1 = 2.464101615 k = 2.464101615 Simplifying k = 2.464101615Subproblem 2
k + 1 = -3.464101615 Simplifying k + 1 = -3.464101615 Reorder the terms: 1 + k = -3.464101615 Solving 1 + k = -3.464101615 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = -3.464101615 + -1 Combine like terms: 1 + -1 = 0 0 + k = -3.464101615 + -1 k = -3.464101615 + -1 Combine like terms: -3.464101615 + -1 = -4.464101615 k = -4.464101615 Simplifying k = -4.464101615Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.464101615, -4.464101615}
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