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Simplifying 7k2 + 13k + 2 = 0 Reorder the terms: 2 + 13k + 7k2 = 0 Solving 2 + 13k + 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'. 0.2857142857 + 1.857142857k + k2 = 0 Move the constant term to the right: Add '-0.2857142857' to each side of the equation. 0.2857142857 + 1.857142857k + -0.2857142857 + k2 = 0 + -0.2857142857 Reorder the terms: 0.2857142857 + -0.2857142857 + 1.857142857k + k2 = 0 + -0.2857142857 Combine like terms: 0.2857142857 + -0.2857142857 = 0.0000000000 0.0000000000 + 1.857142857k + k2 = 0 + -0.2857142857 1.857142857k + k2 = 0 + -0.2857142857 Combine like terms: 0 + -0.2857142857 = -0.2857142857 1.857142857k + k2 = -0.2857142857 The k term is 1.857142857k. Take half its coefficient (0.9285714285). Square it (0.8622448978) and add it to both sides. Add '0.8622448978' to each side of the equation. 1.857142857k + 0.8622448978 + k2 = -0.2857142857 + 0.8622448978 Reorder the terms: 0.8622448978 + 1.857142857k + k2 = -0.2857142857 + 0.8622448978 Combine like terms: -0.2857142857 + 0.8622448978 = 0.5765306121 0.8622448978 + 1.857142857k + k2 = 0.5765306121 Factor a perfect square on the left side: (k + 0.9285714285)(k + 0.9285714285) = 0.5765306121 Calculate the square root of the right side: 0.759296129 Break this problem into two subproblems by setting (k + 0.9285714285) equal to 0.759296129 and -0.759296129.Subproblem 1
k + 0.9285714285 = 0.759296129 Simplifying k + 0.9285714285 = 0.759296129 Reorder the terms: 0.9285714285 + k = 0.759296129 Solving 0.9285714285 + k = 0.759296129 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.9285714285' to each side of the equation. 0.9285714285 + -0.9285714285 + k = 0.759296129 + -0.9285714285 Combine like terms: 0.9285714285 + -0.9285714285 = 0.0000000000 0.0000000000 + k = 0.759296129 + -0.9285714285 k = 0.759296129 + -0.9285714285 Combine like terms: 0.759296129 + -0.9285714285 = -0.1692752995 k = -0.1692752995 Simplifying k = -0.1692752995Subproblem 2
k + 0.9285714285 = -0.759296129 Simplifying k + 0.9285714285 = -0.759296129 Reorder the terms: 0.9285714285 + k = -0.759296129 Solving 0.9285714285 + k = -0.759296129 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.9285714285' to each side of the equation. 0.9285714285 + -0.9285714285 + k = -0.759296129 + -0.9285714285 Combine like terms: 0.9285714285 + -0.9285714285 = 0.0000000000 0.0000000000 + k = -0.759296129 + -0.9285714285 k = -0.759296129 + -0.9285714285 Combine like terms: -0.759296129 + -0.9285714285 = -1.6878675575 k = -1.6878675575 Simplifying k = -1.6878675575Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.1692752995, -1.6878675575}
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