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Simplifying 4k(4) + 32k(3) + 60k(2) = -4k(2)(k + -3)(k + -5) Reorder the terms for easier multiplication: 4 * 4k + 32k(3) + 60k(2) = -4k(2)(k + -3)(k + -5) Multiply 4 * 4 16k + 32k(3) + 60k(2) = -4k(2)(k + -3)(k + -5) Reorder the terms for easier multiplication: 16k + 32 * 3k + 60k(2) = -4k(2)(k + -3)(k + -5) Multiply 32 * 3 16k + 96k + 60k(2) = -4k(2)(k + -3)(k + -5) Reorder the terms for easier multiplication: 16k + 96k + 60 * 2k = -4k(2)(k + -3)(k + -5) Multiply 60 * 2 16k + 96k + 120k = -4k(2)(k + -3)(k + -5) Combine like terms: 16k + 96k = 112k 112k + 120k = -4k(2)(k + -3)(k + -5) Combine like terms: 112k + 120k = 232k 232k = -4k(2)(k + -3)(k + -5) Reorder the terms: 232k = -4k * 2(-3 + k)(k + -5) Reorder the terms: 232k = -4k * 2(-3 + k)(-5 + k) Reorder the terms for easier multiplication: 232k = -4 * 2k(-3 + k)(-5 + k) Multiply -4 * 2 232k = -8k(-3 + k)(-5 + k) Multiply (-3 + k) * (-5 + k) 232k = -8k(-3(-5 + k) + k(-5 + k)) 232k = -8k((-5 * -3 + k * -3) + k(-5 + k)) 232k = -8k((15 + -3k) + k(-5 + k)) 232k = -8k(15 + -3k + (-5 * k + k * k)) 232k = -8k(15 + -3k + (-5k + k2)) Combine like terms: -3k + -5k = -8k 232k = -8k(15 + -8k + k2) 232k = (15 * -8k + -8k * -8k + k2 * -8k) 232k = (-120k + 64k2 + -8k3) Solving 232k = -120k + 64k2 + -8k3 Solving for variable 'k'. Combine like terms: 232k + 120k = 352k 352k + -64k2 + 8k3 = -120k + 64k2 + -8k3 + 120k + -64k2 + 8k3 Reorder the terms: 352k + -64k2 + 8k3 = -120k + 120k + 64k2 + -64k2 + -8k3 + 8k3 Combine like terms: -120k + 120k = 0 352k + -64k2 + 8k3 = 0 + 64k2 + -64k2 + -8k3 + 8k3 352k + -64k2 + 8k3 = 64k2 + -64k2 + -8k3 + 8k3 Combine like terms: 64k2 + -64k2 = 0 352k + -64k2 + 8k3 = 0 + -8k3 + 8k3 352k + -64k2 + 8k3 = -8k3 + 8k3 Combine like terms: -8k3 + 8k3 = 0 352k + -64k2 + 8k3 = 0 Factor out the Greatest Common Factor (GCF), '8k'. 8k(44 + -8k + k2) = 0 Ignore the factor 8.Subproblem 1
Set the factor 'k' equal to zero and attempt to solve: Simplifying k = 0 Solving k = 0 Move all terms containing k to the left, all other terms to the right. Simplifying k = 0Subproblem 2
Set the factor '(44 + -8k + k2)' equal to zero and attempt to solve: Simplifying 44 + -8k + k2 = 0 Solving 44 + -8k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '-44' to each side of the equation. 44 + -8k + -44 + k2 = 0 + -44 Reorder the terms: 44 + -44 + -8k + k2 = 0 + -44 Combine like terms: 44 + -44 = 0 0 + -8k + k2 = 0 + -44 -8k + k2 = 0 + -44 Combine like terms: 0 + -44 = -44 -8k + k2 = -44 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 = -44 + 16 Reorder the terms: 16 + -8k + k2 = -44 + 16 Combine like terms: -44 + 16 = -28 16 + -8k + k2 = -28 Factor a perfect square on the left side: (k + -4)(k + -4) = -28 Can't calculate square root of the right side. The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Solution
k = {0}
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