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Simplifying 7k2 + 41k + 3 = 0 Reorder the terms: 3 + 41k + 7k2 = 0 Solving 3 + 41k + 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'. 0.4285714286 + 5.857142857k + k2 = 0 Move the constant term to the right: Add '-0.4285714286' to each side of the equation. 0.4285714286 + 5.857142857k + -0.4285714286 + k2 = 0 + -0.4285714286 Reorder the terms: 0.4285714286 + -0.4285714286 + 5.857142857k + k2 = 0 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + 5.857142857k + k2 = 0 + -0.4285714286 5.857142857k + k2 = 0 + -0.4285714286 Combine like terms: 0 + -0.4285714286 = -0.4285714286 5.857142857k + k2 = -0.4285714286 The k term is 5.857142857k. Take half its coefficient (2.928571429). Square it (8.576530615) and add it to both sides. Add '8.576530615' to each side of the equation. 5.857142857k + 8.576530615 + k2 = -0.4285714286 + 8.576530615 Reorder the terms: 8.576530615 + 5.857142857k + k2 = -0.4285714286 + 8.576530615 Combine like terms: -0.4285714286 + 8.576530615 = 8.1479591864 8.576530615 + 5.857142857k + k2 = 8.1479591864 Factor a perfect square on the left side: (k + 2.928571429)(k + 2.928571429) = 8.1479591864 Calculate the square root of the right side: 2.854463029 Break this problem into two subproblems by setting (k + 2.928571429) equal to 2.854463029 and -2.854463029.Subproblem 1
k + 2.928571429 = 2.854463029 Simplifying k + 2.928571429 = 2.854463029 Reorder the terms: 2.928571429 + k = 2.854463029 Solving 2.928571429 + k = 2.854463029 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.928571429' to each side of the equation. 2.928571429 + -2.928571429 + k = 2.854463029 + -2.928571429 Combine like terms: 2.928571429 + -2.928571429 = 0.000000000 0.000000000 + k = 2.854463029 + -2.928571429 k = 2.854463029 + -2.928571429 Combine like terms: 2.854463029 + -2.928571429 = -0.0741084 k = -0.0741084 Simplifying k = -0.0741084Subproblem 2
k + 2.928571429 = -2.854463029 Simplifying k + 2.928571429 = -2.854463029 Reorder the terms: 2.928571429 + k = -2.854463029 Solving 2.928571429 + k = -2.854463029 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.928571429' to each side of the equation. 2.928571429 + -2.928571429 + k = -2.854463029 + -2.928571429 Combine like terms: 2.928571429 + -2.928571429 = 0.000000000 0.000000000 + k = -2.854463029 + -2.928571429 k = -2.854463029 + -2.928571429 Combine like terms: -2.854463029 + -2.928571429 = -5.783034458 k = -5.783034458 Simplifying k = -5.783034458Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.0741084, -5.783034458}
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