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Simplifying (k + -1)(k + 1)(k + 3) + 3 = 0 Reorder the terms: (-1 + k)(k + 1)(k + 3) + 3 = 0 Reorder the terms: (-1 + k)(1 + k)(k + 3) + 3 = 0 Reorder the terms: (-1 + k)(1 + k)(3 + k) + 3 = 0 Multiply (-1 + k) * (1 + k) (-1(1 + k) + k(1 + k))(3 + k) + 3 = 0 ((1 * -1 + k * -1) + k(1 + k))(3 + k) + 3 = 0 ((-1 + -1k) + k(1 + k))(3 + k) + 3 = 0 (-1 + -1k + (1 * k + k * k))(3 + k) + 3 = 0 (-1 + -1k + (1k + k2))(3 + k) + 3 = 0 Combine like terms: -1k + 1k = 0 (-1 + 0 + k2)(3 + k) + 3 = 0 (-1 + k2)(3 + k) + 3 = 0 Multiply (-1 + k2) * (3 + k) (-1(3 + k) + k2(3 + k)) + 3 = 0 ((3 * -1 + k * -1) + k2(3 + k)) + 3 = 0 ((-3 + -1k) + k2(3 + k)) + 3 = 0 (-3 + -1k + (3 * k2 + k * k2)) + 3 = 0 (-3 + -1k + (3k2 + k3)) + 3 = 0 (-3 + -1k + 3k2 + k3) + 3 = 0 Reorder the terms: -3 + 3 + -1k + 3k2 + k3 = 0 Combine like terms: -3 + 3 = 0 0 + -1k + 3k2 + k3 = 0 -1k + 3k2 + k3 = 0 Solving -1k + 3k2 + k3 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), 'k'. k(-1 + 3k + k2) = 0Subproblem 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 '(-1 + 3k + k2)' equal to zero and attempt to solve: Simplifying -1 + 3k + k2 = 0 Solving -1 + 3k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + 3k + 1 + k2 = 0 + 1 Reorder the terms: -1 + 1 + 3k + k2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 3k + k2 = 0 + 1 3k + k2 = 0 + 1 Combine like terms: 0 + 1 = 1 3k + k2 = 1 The k term is 3k. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3k + 2.25 + k2 = 1 + 2.25 Reorder the terms: 2.25 + 3k + k2 = 1 + 2.25 Combine like terms: 1 + 2.25 = 3.25 2.25 + 3k + k2 = 3.25 Factor a perfect square on the left side: (k + 1.5)(k + 1.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (k + 1.5) equal to 1.802775638 and -1.802775638.Subproblem 1
k + 1.5 = 1.802775638 Simplifying k + 1.5 = 1.802775638 Reorder the terms: 1.5 + k = 1.802775638 Solving 1.5 + k = 1.802775638 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + k = 1.802775638 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + k = 1.802775638 + -1.5 k = 1.802775638 + -1.5 Combine like terms: 1.802775638 + -1.5 = 0.302775638 k = 0.302775638 Simplifying k = 0.302775638Subproblem 2
k + 1.5 = -1.802775638 Simplifying k + 1.5 = -1.802775638 Reorder the terms: 1.5 + k = -1.802775638 Solving 1.5 + k = -1.802775638 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + k = -1.802775638 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + k = -1.802775638 + -1.5 k = -1.802775638 + -1.5 Combine like terms: -1.802775638 + -1.5 = -3.302775638 k = -3.302775638 Simplifying k = -3.302775638Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.302775638, -3.302775638}Solution
k = {0, 0.302775638, -3.302775638}
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