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Simplifying k(1 + -1k) + -1k(1 + k) + (2k + 1) + -1(1 + -2k) = 11k + 5 (1 * k + -1k * k) + -1k(1 + k) + (2k + 1) + -1(1 + -2k) = 11k + 5 (1k + -1k2) + -1k(1 + k) + (2k + 1) + -1(1 + -2k) = 11k + 5 1k + -1k2 + (1 * -1k + k * -1k) + (2k + 1) + -1(1 + -2k) = 11k + 5 1k + -1k2 + (-1k + -1k2) + (2k + 1) + -1(1 + -2k) = 11k + 5 Reorder the terms: 1k + -1k2 + -1k + -1k2 + (1 + 2k) + -1(1 + -2k) = 11k + 5 Remove parenthesis around (1 + 2k) 1k + -1k2 + -1k + -1k2 + 1 + 2k + -1(1 + -2k) = 11k + 5 1k + -1k2 + -1k + -1k2 + 1 + 2k + (1 * -1 + -2k * -1) = 11k + 5 1k + -1k2 + -1k + -1k2 + 1 + 2k + (-1 + 2k) = 11k + 5 Reorder the terms: 1 + -1 + 1k + -1k + 2k + 2k + -1k2 + -1k2 = 11k + 5 Combine like terms: 1 + -1 = 0 0 + 1k + -1k + 2k + 2k + -1k2 + -1k2 = 11k + 5 1k + -1k + 2k + 2k + -1k2 + -1k2 = 11k + 5 Combine like terms: 1k + -1k = 0 0 + 2k + 2k + -1k2 + -1k2 = 11k + 5 2k + 2k + -1k2 + -1k2 = 11k + 5 Combine like terms: 2k + 2k = 4k 4k + -1k2 + -1k2 = 11k + 5 Combine like terms: -1k2 + -1k2 = -2k2 4k + -2k2 = 11k + 5 Reorder the terms: 4k + -2k2 = 5 + 11k Solving 4k + -2k2 = 5 + 11k Solving for variable 'k'. Reorder the terms: -5 + 4k + -11k + -2k2 = 5 + 11k + -5 + -11k Combine like terms: 4k + -11k = -7k -5 + -7k + -2k2 = 5 + 11k + -5 + -11k Reorder the terms: -5 + -7k + -2k2 = 5 + -5 + 11k + -11k Combine like terms: 5 + -5 = 0 -5 + -7k + -2k2 = 0 + 11k + -11k -5 + -7k + -2k2 = 11k + -11k Combine like terms: 11k + -11k = 0 -5 + -7k + -2k2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(5 + 7k + 2k2) = 0 Factor a trinomial. -1((5 + 2k)(1 + k)) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(5 + 2k)' equal to zero and attempt to solve: Simplifying 5 + 2k = 0 Solving 5 + 2k = 0 Move all terms containing k to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + 2k = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 2k = 0 + -5 2k = 0 + -5 Combine like terms: 0 + -5 = -5 2k = -5 Divide each side by '2'. k = -2.5 Simplifying k = -2.5Subproblem 2
Set the factor '(1 + k)' equal to zero and attempt to solve: Simplifying 1 + k = 0 Solving 1 + k = 0 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 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + k = 0 + -1 k = 0 + -1 Combine like terms: 0 + -1 = -1 k = -1 Simplifying k = -1Solution
k = {-2.5, -1}
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