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Simplifying 458984k + 1 = 35785(k + 56) * 5789k + -478999(60k + -70k) Reorder the terms: 1 + 458984k = 35785(k + 56) * 5789k + -478999(60k + -70k) Reorder the terms: 1 + 458984k = 35785(56 + k) * 5789k + -478999(60k + -70k) Reorder the terms for easier multiplication: 1 + 458984k = 35785 * 5789k(56 + k) + -478999(60k + -70k) Multiply 35785 * 5789 1 + 458984k = 207159365k(56 + k) + -478999(60k + -70k) 1 + 458984k = (56 * 207159365k + k * 207159365k) + -478999(60k + -70k) 1 + 458984k = (11600924440k + 207159365k2) + -478999(60k + -70k) Combine like terms: 60k + -70k = -10k 1 + 458984k = 11600924440k + 207159365k2 + -478999(-10k) Remove parenthesis around (-10k) 1 + 458984k = 11600924440k + 207159365k2 + -478999 * -10k Multiply -478999 * -10 1 + 458984k = 11600924440k + 207159365k2 + 4789990k Reorder the terms: 1 + 458984k = 11600924440k + 4789990k + 207159365k2 Combine like terms: 11600924440k + 4789990k = 11605714430k 1 + 458984k = 11605714430k + 207159365k2 Solving 1 + 458984k = 11605714430k + 207159365k2 Solving for variable 'k'. Combine like terms: 458984k + -11605714430k = -11605255446k 1 + -11605255446k + -207159365k2 = 11605714430k + 207159365k2 + -11605714430k + -207159365k2 Reorder the terms: 1 + -11605255446k + -207159365k2 = 11605714430k + -11605714430k + 207159365k2 + -207159365k2 Combine like terms: 11605714430k + -11605714430k = 0 1 + -11605255446k + -207159365k2 = 0 + 207159365k2 + -207159365k2 1 + -11605255446k + -207159365k2 = 207159365k2 + -207159365k2 Combine like terms: 207159365k2 + -207159365k2 = 0 1 + -11605255446k + -207159365k2 = 0 Begin completing the square. Divide all terms by -207159365 the coefficient of the squared term: Divide each side by '-207159365'. -0.000000004827201512 + 56.02090664k + k2 = 0 Move the constant term to the right: Add '0.000000004827201512' to each side of the equation. -0.000000004827201512 + 56.02090664k + 0.000000004827201512 + k2 = 0 + 0.000000004827201512 Reorder the terms: -0.000000004827201512 + 0.000000004827201512 + 56.02090664k + k2 = 0 + 0.000000004827201512 Combine like terms: -0.000000004827201512 + 0.000000004827201512 = 0.000000000000000000 0.000000000000000000 + 56.02090664k + k2 = 0 + 0.000000004827201512 56.02090664k + k2 = 0 + 0.000000004827201512 Combine like terms: 0 + 0.000000004827201512 = 0.000000004827201512 56.02090664k + k2 = 0.000000004827201512 The k term is 56.02090664k. Take half its coefficient (28.01045332). Square it (784.5854952) and add it to both sides. Add '784.5854952' to each side of the equation. 56.02090664k + 784.5854952 + k2 = 0.000000004827201512 + 784.5854952 Reorder the terms: 784.5854952 + 56.02090664k + k2 = 0.000000004827201512 + 784.5854952 Combine like terms: 0.000000004827201512 + 784.5854952 = 784.585495204827201512 784.5854952 + 56.02090664k + k2 = 784.585495204827201512 Factor a perfect square on the left side: (k + 28.01045332)(k + 28.01045332) = 784.585495204827201512 Calculate the square root of the right side: 28.01045332 Break this problem into two subproblems by setting (k + 28.01045332) equal to 28.01045332 and -28.01045332.Subproblem 1
k + 28.01045332 = 28.01045332 Simplifying k + 28.01045332 = 28.01045332 Reorder the terms: 28.01045332 + k = 28.01045332 Add '-28.01045332' to each side of the equation. 28.01045332 + -28.01045332 + k = 28.01045332 + -28.01045332 Combine like terms: 28.01045332 + -28.01045332 = 0.00000000 0.00000000 + k = 28.01045332 + -28.01045332 k = 28.01045332 + -28.01045332 Combine like terms: 28.01045332 + -28.01045332 = 0.00000000 k = 0.00000000 Solving k = 0.00000000 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Simplifying k = 0.00000000Subproblem 2
k + 28.01045332 = -28.01045332 Simplifying k + 28.01045332 = -28.01045332 Reorder the terms: 28.01045332 + k = -28.01045332 Solving 28.01045332 + k = -28.01045332 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-28.01045332' to each side of the equation. 28.01045332 + -28.01045332 + k = -28.01045332 + -28.01045332 Combine like terms: 28.01045332 + -28.01045332 = 0.00000000 0.00000000 + k = -28.01045332 + -28.01045332 k = -28.01045332 + -28.01045332 Combine like terms: -28.01045332 + -28.01045332 = -56.02090664 k = -56.02090664 Simplifying k = -56.02090664Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.00000000, -56.02090664}
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