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Simplifying (k + -6)(k + -2) = 3 Reorder the terms: (-6 + k)(k + -2) = 3 Reorder the terms: (-6 + k)(-2 + k) = 3 Multiply (-6 + k) * (-2 + k) (-6(-2 + k) + k(-2 + k)) = 3 ((-2 * -6 + k * -6) + k(-2 + k)) = 3 ((12 + -6k) + k(-2 + k)) = 3 (12 + -6k + (-2 * k + k * k)) = 3 (12 + -6k + (-2k + k2)) = 3 Combine like terms: -6k + -2k = -8k (12 + -8k + k2) = 3 Solving 12 + -8k + k2 = 3 Solving for variable 'k'. Reorder the terms: 12 + -3 + -8k + k2 = 3 + -3 Combine like terms: 12 + -3 = 9 9 + -8k + k2 = 3 + -3 Combine like terms: 3 + -3 = 0 9 + -8k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '-9' to each side of the equation. 9 + -8k + -9 + k2 = 0 + -9 Reorder the terms: 9 + -9 + -8k + k2 = 0 + -9 Combine like terms: 9 + -9 = 0 0 + -8k + k2 = 0 + -9 -8k + k2 = 0 + -9 Combine like terms: 0 + -9 = -9 -8k + k2 = -9 The k term is -8k. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8k + 16 + k2 = -9 + 16 Reorder the terms: 16 + -8k + k2 = -9 + 16 Combine like terms: -9 + 16 = 7 16 + -8k + k2 = 7 Factor a perfect square on the left side: (k + -4)(k + -4) = 7 Calculate the square root of the right side: 2.645751311 Break this problem into two subproblems by setting (k + -4) equal to 2.645751311 and -2.645751311.Subproblem 1
k + -4 = 2.645751311 Simplifying k + -4 = 2.645751311 Reorder the terms: -4 + k = 2.645751311 Solving -4 + k = 2.645751311 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + k = 2.645751311 + 4 Combine like terms: -4 + 4 = 0 0 + k = 2.645751311 + 4 k = 2.645751311 + 4 Combine like terms: 2.645751311 + 4 = 6.645751311 k = 6.645751311 Simplifying k = 6.645751311Subproblem 2
k + -4 = -2.645751311 Simplifying k + -4 = -2.645751311 Reorder the terms: -4 + k = -2.645751311 Solving -4 + k = -2.645751311 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + k = -2.645751311 + 4 Combine like terms: -4 + 4 = 0 0 + k = -2.645751311 + 4 k = -2.645751311 + 4 Combine like terms: -2.645751311 + 4 = 1.354248689 k = 1.354248689 Simplifying k = 1.354248689Solution
The solution to the problem is based on the solutions from the subproblems. k = {6.645751311, 1.354248689}
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