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Simplifying 21k2 + 4k + -4 = 0 Reorder the terms: -4 + 4k + 21k2 = 0 Solving -4 + 4k + 21k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -0.1904761905 + 0.1904761905k + k2 = 0 Move the constant term to the right: Add '0.1904761905' to each side of the equation. -0.1904761905 + 0.1904761905k + 0.1904761905 + k2 = 0 + 0.1904761905 Reorder the terms: -0.1904761905 + 0.1904761905 + 0.1904761905k + k2 = 0 + 0.1904761905 Combine like terms: -0.1904761905 + 0.1904761905 = 0.0000000000 0.0000000000 + 0.1904761905k + k2 = 0 + 0.1904761905 0.1904761905k + k2 = 0 + 0.1904761905 Combine like terms: 0 + 0.1904761905 = 0.1904761905 0.1904761905k + k2 = 0.1904761905 The k term is 0.1904761905k. Take half its coefficient (0.09523809525). Square it (0.009070294787) and add it to both sides. Add '0.009070294787' to each side of the equation. 0.1904761905k + 0.009070294787 + k2 = 0.1904761905 + 0.009070294787 Reorder the terms: 0.009070294787 + 0.1904761905k + k2 = 0.1904761905 + 0.009070294787 Combine like terms: 0.1904761905 + 0.009070294787 = 0.199546485287 0.009070294787 + 0.1904761905k + k2 = 0.199546485287 Factor a perfect square on the left side: (k + 0.09523809525)(k + 0.09523809525) = 0.199546485287 Calculate the square root of the right side: 0.446706263 Break this problem into two subproblems by setting (k + 0.09523809525) equal to 0.446706263 and -0.446706263.Subproblem 1
k + 0.09523809525 = 0.446706263 Simplifying k + 0.09523809525 = 0.446706263 Reorder the terms: 0.09523809525 + k = 0.446706263 Solving 0.09523809525 + k = 0.446706263 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.09523809525' to each side of the equation. 0.09523809525 + -0.09523809525 + k = 0.446706263 + -0.09523809525 Combine like terms: 0.09523809525 + -0.09523809525 = 0.00000000000 0.00000000000 + k = 0.446706263 + -0.09523809525 k = 0.446706263 + -0.09523809525 Combine like terms: 0.446706263 + -0.09523809525 = 0.35146816775 k = 0.35146816775 Simplifying k = 0.35146816775Subproblem 2
k + 0.09523809525 = -0.446706263 Simplifying k + 0.09523809525 = -0.446706263 Reorder the terms: 0.09523809525 + k = -0.446706263 Solving 0.09523809525 + k = -0.446706263 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.09523809525' to each side of the equation. 0.09523809525 + -0.09523809525 + k = -0.446706263 + -0.09523809525 Combine like terms: 0.09523809525 + -0.09523809525 = 0.00000000000 0.00000000000 + k = -0.446706263 + -0.09523809525 k = -0.446706263 + -0.09523809525 Combine like terms: -0.446706263 + -0.09523809525 = -0.54194435825 k = -0.54194435825 Simplifying k = -0.54194435825Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.35146816775, -0.54194435825}
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