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Simplifying 7k2 + -16k + -100 = 0 Reorder the terms: -100 + -16k + 7k2 = 0 Solving -100 + -16k + 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'. -14.28571429 + -2.285714286k + k2 = 0 Move the constant term to the right: Add '14.28571429' to each side of the equation. -14.28571429 + -2.285714286k + 14.28571429 + k2 = 0 + 14.28571429 Reorder the terms: -14.28571429 + 14.28571429 + -2.285714286k + k2 = 0 + 14.28571429 Combine like terms: -14.28571429 + 14.28571429 = 0.00000000 0.00000000 + -2.285714286k + k2 = 0 + 14.28571429 -2.285714286k + k2 = 0 + 14.28571429 Combine like terms: 0 + 14.28571429 = 14.28571429 -2.285714286k + k2 = 14.28571429 The k term is -2.285714286k. Take half its coefficient (-1.142857143). Square it (1.306122449) and add it to both sides. Add '1.306122449' to each side of the equation. -2.285714286k + 1.306122449 + k2 = 14.28571429 + 1.306122449 Reorder the terms: 1.306122449 + -2.285714286k + k2 = 14.28571429 + 1.306122449 Combine like terms: 14.28571429 + 1.306122449 = 15.591836739 1.306122449 + -2.285714286k + k2 = 15.591836739 Factor a perfect square on the left side: (k + -1.142857143)(k + -1.142857143) = 15.591836739 Calculate the square root of the right side: 3.948649989 Break this problem into two subproblems by setting (k + -1.142857143) equal to 3.948649989 and -3.948649989.Subproblem 1
k + -1.142857143 = 3.948649989 Simplifying k + -1.142857143 = 3.948649989 Reorder the terms: -1.142857143 + k = 3.948649989 Solving -1.142857143 + k = 3.948649989 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + k = 3.948649989 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + k = 3.948649989 + 1.142857143 k = 3.948649989 + 1.142857143 Combine like terms: 3.948649989 + 1.142857143 = 5.091507132 k = 5.091507132 Simplifying k = 5.091507132Subproblem 2
k + -1.142857143 = -3.948649989 Simplifying k + -1.142857143 = -3.948649989 Reorder the terms: -1.142857143 + k = -3.948649989 Solving -1.142857143 + k = -3.948649989 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + k = -3.948649989 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + k = -3.948649989 + 1.142857143 k = -3.948649989 + 1.142857143 Combine like terms: -3.948649989 + 1.142857143 = -2.805792846 k = -2.805792846 Simplifying k = -2.805792846Solution
The solution to the problem is based on the solutions from the subproblems. k = {5.091507132, -2.805792846}
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