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Simplifying 4k + k(-1k + -1) = -5(1 + -2k) Reorder the terms: 4k + k(-1 + -1k) = -5(1 + -2k) 4k + (-1 * k + -1k * k) = -5(1 + -2k) 4k + (-1k + -1k2) = -5(1 + -2k) Combine like terms: 4k + -1k = 3k 3k + -1k2 = -5(1 + -2k) 3k + -1k2 = (1 * -5 + -2k * -5) 3k + -1k2 = (-5 + 10k) Solving 3k + -1k2 = -5 + 10k Solving for variable 'k'. Reorder the terms: 5 + 3k + -10k + -1k2 = -5 + 10k + 5 + -10k Combine like terms: 3k + -10k = -7k 5 + -7k + -1k2 = -5 + 10k + 5 + -10k Reorder the terms: 5 + -7k + -1k2 = -5 + 5 + 10k + -10k Combine like terms: -5 + 5 = 0 5 + -7k + -1k2 = 0 + 10k + -10k 5 + -7k + -1k2 = 10k + -10k Combine like terms: 10k + -10k = 0 5 + -7k + -1k2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -5 + 7k + k2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 7k + 5 + k2 = 0 + 5 Reorder the terms: -5 + 5 + 7k + k2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 7k + k2 = 0 + 5 7k + k2 = 0 + 5 Combine like terms: 0 + 5 = 5 7k + k2 = 5 The k term is 7k. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7k + 12.25 + k2 = 5 + 12.25 Reorder the terms: 12.25 + 7k + k2 = 5 + 12.25 Combine like terms: 5 + 12.25 = 17.25 12.25 + 7k + k2 = 17.25 Factor a perfect square on the left side: (k + 3.5)(k + 3.5) = 17.25 Calculate the square root of the right side: 4.153311931 Break this problem into two subproblems by setting (k + 3.5) equal to 4.153311931 and -4.153311931.Subproblem 1
k + 3.5 = 4.153311931 Simplifying k + 3.5 = 4.153311931 Reorder the terms: 3.5 + k = 4.153311931 Solving 3.5 + k = 4.153311931 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + k = 4.153311931 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + k = 4.153311931 + -3.5 k = 4.153311931 + -3.5 Combine like terms: 4.153311931 + -3.5 = 0.653311931 k = 0.653311931 Simplifying k = 0.653311931Subproblem 2
k + 3.5 = -4.153311931 Simplifying k + 3.5 = -4.153311931 Reorder the terms: 3.5 + k = -4.153311931 Solving 3.5 + k = -4.153311931 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + k = -4.153311931 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + k = -4.153311931 + -3.5 k = -4.153311931 + -3.5 Combine like terms: -4.153311931 + -3.5 = -7.653311931 k = -7.653311931 Simplifying k = -7.653311931Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.653311931, -7.653311931}
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