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Simplifying -4k2 + -8k + -3 = -3 + -5k3 Reorder the terms: -3 + -8k + -4k2 = -3 + -5k3 Add '3' to each side of the equation. -3 + -8k + 3 + -4k2 = -3 + 3 + -5k3 Reorder the terms: -3 + 3 + -8k + -4k2 = -3 + 3 + -5k3 Combine like terms: -3 + 3 = 0 0 + -8k + -4k2 = -3 + 3 + -5k3 -8k + -4k2 = -3 + 3 + -5k3 Combine like terms: -3 + 3 = 0 -8k + -4k2 = 0 + -5k3 -8k + -4k2 = -5k3 Solving -8k + -4k2 = -5k3 Solving for variable 'k'. Combine like terms: -5k3 + 5k3 = 0 -8k + -4k2 + 5k3 = 0 Factor out the Greatest Common Factor (GCF), 'k'. k(-8 + -4k + 5k2) = 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 '(-8 + -4k + 5k2)' equal to zero and attempt to solve: Simplifying -8 + -4k + 5k2 = 0 Solving -8 + -4k + 5k2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1.6 + -0.8k + k2 = 0 Move the constant term to the right: Add '1.6' to each side of the equation. -1.6 + -0.8k + 1.6 + k2 = 0 + 1.6 Reorder the terms: -1.6 + 1.6 + -0.8k + k2 = 0 + 1.6 Combine like terms: -1.6 + 1.6 = 0.0 0.0 + -0.8k + k2 = 0 + 1.6 -0.8k + k2 = 0 + 1.6 Combine like terms: 0 + 1.6 = 1.6 -0.8k + k2 = 1.6 The k term is -0.8k. Take half its coefficient (-0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. -0.8k + 0.16 + k2 = 1.6 + 0.16 Reorder the terms: 0.16 + -0.8k + k2 = 1.6 + 0.16 Combine like terms: 1.6 + 0.16 = 1.76 0.16 + -0.8k + k2 = 1.76 Factor a perfect square on the left side: (k + -0.4)(k + -0.4) = 1.76 Calculate the square root of the right side: 1.326649916 Break this problem into two subproblems by setting (k + -0.4) equal to 1.326649916 and -1.326649916.Subproblem 1
k + -0.4 = 1.326649916 Simplifying k + -0.4 = 1.326649916 Reorder the terms: -0.4 + k = 1.326649916 Solving -0.4 + k = 1.326649916 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + k = 1.326649916 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + k = 1.326649916 + 0.4 k = 1.326649916 + 0.4 Combine like terms: 1.326649916 + 0.4 = 1.726649916 k = 1.726649916 Simplifying k = 1.726649916Subproblem 2
k + -0.4 = -1.326649916 Simplifying k + -0.4 = -1.326649916 Reorder the terms: -0.4 + k = -1.326649916 Solving -0.4 + k = -1.326649916 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + k = -1.326649916 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + k = -1.326649916 + 0.4 k = -1.326649916 + 0.4 Combine like terms: -1.326649916 + 0.4 = -0.926649916 k = -0.926649916 Simplifying k = -0.926649916Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.726649916, -0.926649916}Solution
k = {0, 1.726649916, -0.926649916}
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