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Simplifying 7k2 + 29k + -37 = 0 Reorder the terms: -37 + 29k + 7k2 = 0 Solving -37 + 29k + 7k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -5.285714286 + 4.142857143k + k2 = 0 Move the constant term to the right: Add '5.285714286' to each side of the equation. -5.285714286 + 4.142857143k + 5.285714286 + k2 = 0 + 5.285714286 Reorder the terms: -5.285714286 + 5.285714286 + 4.142857143k + k2 = 0 + 5.285714286 Combine like terms: -5.285714286 + 5.285714286 = 0.000000000 0.000000000 + 4.142857143k + k2 = 0 + 5.285714286 4.142857143k + k2 = 0 + 5.285714286 Combine like terms: 0 + 5.285714286 = 5.285714286 4.142857143k + k2 = 5.285714286 The k term is 4.142857143k. Take half its coefficient (2.071428572). Square it (4.290816329) and add it to both sides. Add '4.290816329' to each side of the equation. 4.142857143k + 4.290816329 + k2 = 5.285714286 + 4.290816329 Reorder the terms: 4.290816329 + 4.142857143k + k2 = 5.285714286 + 4.290816329 Combine like terms: 5.285714286 + 4.290816329 = 9.576530615 4.290816329 + 4.142857143k + k2 = 9.576530615 Factor a perfect square on the left side: (k + 2.071428572)(k + 2.071428572) = 9.576530615 Calculate the square root of the right side: 3.094597004 Break this problem into two subproblems by setting (k + 2.071428572) equal to 3.094597004 and -3.094597004.Subproblem 1
k + 2.071428572 = 3.094597004 Simplifying k + 2.071428572 = 3.094597004 Reorder the terms: 2.071428572 + k = 3.094597004 Solving 2.071428572 + k = 3.094597004 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.071428572' to each side of the equation. 2.071428572 + -2.071428572 + k = 3.094597004 + -2.071428572 Combine like terms: 2.071428572 + -2.071428572 = 0.000000000 0.000000000 + k = 3.094597004 + -2.071428572 k = 3.094597004 + -2.071428572 Combine like terms: 3.094597004 + -2.071428572 = 1.023168432 k = 1.023168432 Simplifying k = 1.023168432Subproblem 2
k + 2.071428572 = -3.094597004 Simplifying k + 2.071428572 = -3.094597004 Reorder the terms: 2.071428572 + k = -3.094597004 Solving 2.071428572 + k = -3.094597004 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.071428572' to each side of the equation. 2.071428572 + -2.071428572 + k = -3.094597004 + -2.071428572 Combine like terms: 2.071428572 + -2.071428572 = 0.000000000 0.000000000 + k = -3.094597004 + -2.071428572 k = -3.094597004 + -2.071428572 Combine like terms: -3.094597004 + -2.071428572 = -5.166025576 k = -5.166025576 Simplifying k = -5.166025576Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.023168432, -5.166025576}
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