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Simplifying (2x + -1k)(3x + 2k) = 6x2 + kx + -32 Reorder the terms: (-1k + 2x)(3x + 2k) = 6x2 + kx + -32 Reorder the terms: (-1k + 2x)(2k + 3x) = 6x2 + kx + -32 Multiply (-1k + 2x) * (2k + 3x) (-1k * (2k + 3x) + 2x * (2k + 3x)) = 6x2 + kx + -32 ((2k * -1k + 3x * -1k) + 2x * (2k + 3x)) = 6x2 + kx + -32 Reorder the terms: ((-3kx + -2k2) + 2x * (2k + 3x)) = 6x2 + kx + -32 ((-3kx + -2k2) + 2x * (2k + 3x)) = 6x2 + kx + -32 (-3kx + -2k2 + (2k * 2x + 3x * 2x)) = 6x2 + kx + -32 (-3kx + -2k2 + (4kx + 6x2)) = 6x2 + kx + -32 Reorder the terms: (-3kx + 4kx + -2k2 + 6x2) = 6x2 + kx + -32 Combine like terms: -3kx + 4kx = 1kx (1kx + -2k2 + 6x2) = 6x2 + kx + -32 Reorder the terms: 1kx + -2k2 + 6x2 = -32 + kx + 6x2 Add '-6x2' to each side of the equation. 1kx + -2k2 + 6x2 + -6x2 = -32 + kx + 6x2 + -6x2 Combine like terms: 6x2 + -6x2 = 0 1kx + -2k2 + 0 = -32 + kx + 6x2 + -6x2 1kx + -2k2 = -32 + kx + 6x2 + -6x2 Combine like terms: 6x2 + -6x2 = 0 1kx + -2k2 = -32 + kx + 0 1kx + -2k2 = -32 + kx Solving 1kx + -2k2 = -32 + kx Solving for variable 'k'. Reorder the terms: 32 + 1kx + -1kx + -2k2 = -32 + kx + 32 + -1kx Combine like terms: 1kx + -1kx = 0 32 + 0 + -2k2 = -32 + kx + 32 + -1kx 32 + -2k2 = -32 + kx + 32 + -1kx Reorder the terms: 32 + -2k2 = -32 + 32 + kx + -1kx Combine like terms: -32 + 32 = 0 32 + -2k2 = 0 + kx + -1kx 32 + -2k2 = kx + -1kx Combine like terms: kx + -1kx = 0 32 + -2k2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(16 + -1k2) = 0 Factor a difference between two squares. 2((4 + k)(4 + -1k)) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(4 + k)' equal to zero and attempt to solve: Simplifying 4 + k = 0 Solving 4 + k = 0 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 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + k = 0 + -4 k = 0 + -4 Combine like terms: 0 + -4 = -4 k = -4 Simplifying k = -4Subproblem 2
Set the factor '(4 + -1k)' equal to zero and attempt to solve: Simplifying 4 + -1k = 0 Solving 4 + -1k = 0 Move all terms containing k to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + -1k = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -1k = 0 + -4 -1k = 0 + -4 Combine like terms: 0 + -4 = -4 -1k = -4 Divide each side by '-1'. k = 4 Simplifying k = 4Solution
k = {-4, 4}
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