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Simplifying 7k2 + 30k + -23 = 0 Reorder the terms: -23 + 30k + 7k2 = 0 Solving -23 + 30k + 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'. -3.285714286 + 4.285714286k + k2 = 0 Move the constant term to the right: Add '3.285714286' to each side of the equation. -3.285714286 + 4.285714286k + 3.285714286 + k2 = 0 + 3.285714286 Reorder the terms: -3.285714286 + 3.285714286 + 4.285714286k + k2 = 0 + 3.285714286 Combine like terms: -3.285714286 + 3.285714286 = 0.000000000 0.000000000 + 4.285714286k + k2 = 0 + 3.285714286 4.285714286k + k2 = 0 + 3.285714286 Combine like terms: 0 + 3.285714286 = 3.285714286 4.285714286k + k2 = 3.285714286 The k term is 4.285714286k. Take half its coefficient (2.142857143). Square it (4.591836735) and add it to both sides. Add '4.591836735' to each side of the equation. 4.285714286k + 4.591836735 + k2 = 3.285714286 + 4.591836735 Reorder the terms: 4.591836735 + 4.285714286k + k2 = 3.285714286 + 4.591836735 Combine like terms: 3.285714286 + 4.591836735 = 7.877551021 4.591836735 + 4.285714286k + k2 = 7.877551021 Factor a perfect square on the left side: (k + 2.142857143)(k + 2.142857143) = 7.877551021 Calculate the square root of the right side: 2.806697529 Break this problem into two subproblems by setting (k + 2.142857143) equal to 2.806697529 and -2.806697529.Subproblem 1
k + 2.142857143 = 2.806697529 Simplifying k + 2.142857143 = 2.806697529 Reorder the terms: 2.142857143 + k = 2.806697529 Solving 2.142857143 + k = 2.806697529 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.142857143' to each side of the equation. 2.142857143 + -2.142857143 + k = 2.806697529 + -2.142857143 Combine like terms: 2.142857143 + -2.142857143 = 0.000000000 0.000000000 + k = 2.806697529 + -2.142857143 k = 2.806697529 + -2.142857143 Combine like terms: 2.806697529 + -2.142857143 = 0.663840386 k = 0.663840386 Simplifying k = 0.663840386Subproblem 2
k + 2.142857143 = -2.806697529 Simplifying k + 2.142857143 = -2.806697529 Reorder the terms: 2.142857143 + k = -2.806697529 Solving 2.142857143 + k = -2.806697529 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.142857143' to each side of the equation. 2.142857143 + -2.142857143 + k = -2.806697529 + -2.142857143 Combine like terms: 2.142857143 + -2.142857143 = 0.000000000 0.000000000 + k = -2.806697529 + -2.142857143 k = -2.806697529 + -2.142857143 Combine like terms: -2.806697529 + -2.142857143 = -4.949554672 k = -4.949554672 Simplifying k = -4.949554672Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.663840386, -4.949554672}
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