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Simplifying 4k2 + 20k + -5 = 0 Reorder the terms: -5 + 20k + 4k2 = 0 Solving -5 + 20k + 4k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.25 + 5k + k2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + 5k + 1.25 + k2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + 5k + k2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + 5k + k2 = 0 + 1.25 5k + k2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 5k + k2 = 1.25 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 = 1.25 + 6.25 Reorder the terms: 6.25 + 5k + k2 = 1.25 + 6.25 Combine like terms: 1.25 + 6.25 = 7.5 6.25 + 5k + k2 = 7.5 Factor a perfect square on the left side: (k + 2.5)(k + 2.5) = 7.5 Calculate the square root of the right side: 2.738612788 Break this problem into two subproblems by setting (k + 2.5) equal to 2.738612788 and -2.738612788.Subproblem 1
k + 2.5 = 2.738612788 Simplifying k + 2.5 = 2.738612788 Reorder the terms: 2.5 + k = 2.738612788 Solving 2.5 + k = 2.738612788 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 = 2.738612788 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + k = 2.738612788 + -2.5 k = 2.738612788 + -2.5 Combine like terms: 2.738612788 + -2.5 = 0.238612788 k = 0.238612788 Simplifying k = 0.238612788Subproblem 2
k + 2.5 = -2.738612788 Simplifying k + 2.5 = -2.738612788 Reorder the terms: 2.5 + k = -2.738612788 Solving 2.5 + k = -2.738612788 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 = -2.738612788 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + k = -2.738612788 + -2.5 k = -2.738612788 + -2.5 Combine like terms: -2.738612788 + -2.5 = -5.238612788 k = -5.238612788 Simplifying k = -5.238612788Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.238612788, -5.238612788}
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