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Simplifying 4k2 + 40k = 10 Reorder the terms: 40k + 4k2 = 10 Solving 40k + 4k2 = 10 Solving for variable 'k'. Reorder the terms: -10 + 40k + 4k2 = 10 + -10 Combine like terms: 10 + -10 = 0 -10 + 40k + 4k2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + 20k + 2k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + 20k + 2k2)' equal to zero and attempt to solve: Simplifying -5 + 20k + 2k2 = 0 Solving -5 + 20k + 2k2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2.5 + 10k + k2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 10k + 2.5 + k2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 10k + k2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 10k + k2 = 0 + 2.5 10k + k2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 10k + k2 = 2.5 The k term is 10k. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10k + 25 + k2 = 2.5 + 25 Reorder the terms: 25 + 10k + k2 = 2.5 + 25 Combine like terms: 2.5 + 25 = 27.5 25 + 10k + k2 = 27.5 Factor a perfect square on the left side: (k + 5)(k + 5) = 27.5 Calculate the square root of the right side: 5.244044241 Break this problem into two subproblems by setting (k + 5) equal to 5.244044241 and -5.244044241.Subproblem 1
k + 5 = 5.244044241 Simplifying k + 5 = 5.244044241 Reorder the terms: 5 + k = 5.244044241 Solving 5 + k = 5.244044241 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + k = 5.244044241 + -5 Combine like terms: 5 + -5 = 0 0 + k = 5.244044241 + -5 k = 5.244044241 + -5 Combine like terms: 5.244044241 + -5 = 0.244044241 k = 0.244044241 Simplifying k = 0.244044241Subproblem 2
k + 5 = -5.244044241 Simplifying k + 5 = -5.244044241 Reorder the terms: 5 + k = -5.244044241 Solving 5 + k = -5.244044241 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + k = -5.244044241 + -5 Combine like terms: 5 + -5 = 0 0 + k = -5.244044241 + -5 k = -5.244044241 + -5 Combine like terms: -5.244044241 + -5 = -10.244044241 k = -10.244044241 Simplifying k = -10.244044241Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.244044241, -10.244044241}Solution
k = {0.244044241, -10.244044241}
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