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Simplifying 4k2 + -20k + 1 = 0 Reorder the terms: 1 + -20k + 4k2 = 0 Solving 1 + -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'. 0.25 + -5k + k2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -5k + -0.25 + k2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -5k + k2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -5k + k2 = 0 + -0.25 -5k + k2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -5k + k2 = -0.25 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 = -0.25 + 6.25 Reorder the terms: 6.25 + -5k + k2 = -0.25 + 6.25 Combine like terms: -0.25 + 6.25 = 6 6.25 + -5k + k2 = 6 Factor a perfect square on the left side: (k + -2.5)(k + -2.5) = 6 Calculate the square root of the right side: 2.449489743 Break this problem into two subproblems by setting (k + -2.5) equal to 2.449489743 and -2.449489743.Subproblem 1
k + -2.5 = 2.449489743 Simplifying k + -2.5 = 2.449489743 Reorder the terms: -2.5 + k = 2.449489743 Solving -2.5 + k = 2.449489743 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 = 2.449489743 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + k = 2.449489743 + 2.5 k = 2.449489743 + 2.5 Combine like terms: 2.449489743 + 2.5 = 4.949489743 k = 4.949489743 Simplifying k = 4.949489743Subproblem 2
k + -2.5 = -2.449489743 Simplifying k + -2.5 = -2.449489743 Reorder the terms: -2.5 + k = -2.449489743 Solving -2.5 + k = -2.449489743 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 = -2.449489743 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + k = -2.449489743 + 2.5 k = -2.449489743 + 2.5 Combine like terms: -2.449489743 + 2.5 = 0.050510257 k = 0.050510257 Simplifying k = 0.050510257Solution
The solution to the problem is based on the solutions from the subproblems. k = {4.949489743, 0.050510257}
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