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Simplifying 7k(k + -3) = 3k + -5 Reorder the terms: 7k(-3 + k) = 3k + -5 (-3 * 7k + k * 7k) = 3k + -5 (-21k + 7k2) = 3k + -5 Reorder the terms: -21k + 7k2 = -5 + 3k Solving -21k + 7k2 = -5 + 3k Solving for variable 'k'. Reorder the terms: 5 + -21k + -3k + 7k2 = -5 + 3k + 5 + -3k Combine like terms: -21k + -3k = -24k 5 + -24k + 7k2 = -5 + 3k + 5 + -3k Reorder the terms: 5 + -24k + 7k2 = -5 + 5 + 3k + -3k Combine like terms: -5 + 5 = 0 5 + -24k + 7k2 = 0 + 3k + -3k 5 + -24k + 7k2 = 3k + -3k Combine like terms: 3k + -3k = 0 5 + -24k + 7k2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 0.7142857143 + -3.428571429k + k2 = 0 Move the constant term to the right: Add '-0.7142857143' to each side of the equation. 0.7142857143 + -3.428571429k + -0.7142857143 + k2 = 0 + -0.7142857143 Reorder the terms: 0.7142857143 + -0.7142857143 + -3.428571429k + k2 = 0 + -0.7142857143 Combine like terms: 0.7142857143 + -0.7142857143 = 0.0000000000 0.0000000000 + -3.428571429k + k2 = 0 + -0.7142857143 -3.428571429k + k2 = 0 + -0.7142857143 Combine like terms: 0 + -0.7142857143 = -0.7142857143 -3.428571429k + k2 = -0.7142857143 The k term is -3.428571429k. Take half its coefficient (-1.714285715). Square it (2.938775513) and add it to both sides. Add '2.938775513' to each side of the equation. -3.428571429k + 2.938775513 + k2 = -0.7142857143 + 2.938775513 Reorder the terms: 2.938775513 + -3.428571429k + k2 = -0.7142857143 + 2.938775513 Combine like terms: -0.7142857143 + 2.938775513 = 2.2244897987 2.938775513 + -3.428571429k + k2 = 2.2244897987 Factor a perfect square on the left side: (k + -1.714285715)(k + -1.714285715) = 2.2244897987 Calculate the square root of the right side: 1.491472359 Break this problem into two subproblems by setting (k + -1.714285715) equal to 1.491472359 and -1.491472359.Subproblem 1
k + -1.714285715 = 1.491472359 Simplifying k + -1.714285715 = 1.491472359 Reorder the terms: -1.714285715 + k = 1.491472359 Solving -1.714285715 + k = 1.491472359 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.714285715' to each side of the equation. -1.714285715 + 1.714285715 + k = 1.491472359 + 1.714285715 Combine like terms: -1.714285715 + 1.714285715 = 0.000000000 0.000000000 + k = 1.491472359 + 1.714285715 k = 1.491472359 + 1.714285715 Combine like terms: 1.491472359 + 1.714285715 = 3.205758074 k = 3.205758074 Simplifying k = 3.205758074Subproblem 2
k + -1.714285715 = -1.491472359 Simplifying k + -1.714285715 = -1.491472359 Reorder the terms: -1.714285715 + k = -1.491472359 Solving -1.714285715 + k = -1.491472359 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.714285715' to each side of the equation. -1.714285715 + 1.714285715 + k = -1.491472359 + 1.714285715 Combine like terms: -1.714285715 + 1.714285715 = 0.000000000 0.000000000 + k = -1.491472359 + 1.714285715 k = -1.491472359 + 1.714285715 Combine like terms: -1.491472359 + 1.714285715 = 0.222813356 k = 0.222813356 Simplifying k = 0.222813356Solution
The solution to the problem is based on the solutions from the subproblems. k = {3.205758074, 0.222813356}
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