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Simplifying 4k2 + 20k + -2 = 0 Reorder the terms: -2 + 20k + 4k2 = 0 Solving -2 + 20k + 4k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + 10k + 2k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + 10k + 2k2)' equal to zero and attempt to solve: Simplifying -1 + 10k + 2k2 = 0 Solving -1 + 10k + 2k2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + 5k + k2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 5k + 0.5 + k2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 5k + k2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 5k + k2 = 0 + 0.5 5k + k2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 5k + k2 = 0.5 The k term is 5k. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5k + 6.25 + k2 = 0.5 + 6.25 Reorder the terms: 6.25 + 5k + k2 = 0.5 + 6.25 Combine like terms: 0.5 + 6.25 = 6.75 6.25 + 5k + k2 = 6.75 Factor a perfect square on the left side: (k + 2.5)(k + 2.5) = 6.75 Calculate the square root of the right side: 2.598076211 Break this problem into two subproblems by setting (k + 2.5) equal to 2.598076211 and -2.598076211.Subproblem 1
k + 2.5 = 2.598076211 Simplifying k + 2.5 = 2.598076211 Reorder the terms: 2.5 + k = 2.598076211 Solving 2.5 + k = 2.598076211 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + k = 2.598076211 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + k = 2.598076211 + -2.5 k = 2.598076211 + -2.5 Combine like terms: 2.598076211 + -2.5 = 0.098076211 k = 0.098076211 Simplifying k = 0.098076211Subproblem 2
k + 2.5 = -2.598076211 Simplifying k + 2.5 = -2.598076211 Reorder the terms: 2.5 + k = -2.598076211 Solving 2.5 + k = -2.598076211 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + k = -2.598076211 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + k = -2.598076211 + -2.5 k = -2.598076211 + -2.5 Combine like terms: -2.598076211 + -2.5 = -5.098076211 k = -5.098076211 Simplifying k = -5.098076211Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.098076211, -5.098076211}Solution
k = {0.098076211, -5.098076211}
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