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Simplifying 34.8k + 0.2k(k + -4) = 1.2 + -9(2 + -3k) Reorder the terms: 34.8k + 0.2k(-4 + k) = 1.2 + -9(2 + -3k) 34.8k + (-4 * 0.2k + k * 0.2k) = 1.2 + -9(2 + -3k) 34.8k + (-0.8k + 0.2k2) = 1.2 + -9(2 + -3k) Combine like terms: 34.8k + -0.8k = 34k 34k + 0.2k2 = 1.2 + -9(2 + -3k) 34k + 0.2k2 = 1.2 + (2 * -9 + -3k * -9) 34k + 0.2k2 = 1.2 + (-18 + 27k) Combine like terms: 1.2 + -18 = -16.8 34k + 0.2k2 = -16.8 + 27k Solving 34k + 0.2k2 = -16.8 + 27k Solving for variable 'k'. Reorder the terms: 16.8 + 34k + -27k + 0.2k2 = -16.8 + 27k + 16.8 + -27k Combine like terms: 34k + -27k = 7k 16.8 + 7k + 0.2k2 = -16.8 + 27k + 16.8 + -27k Reorder the terms: 16.8 + 7k + 0.2k2 = -16.8 + 16.8 + 27k + -27k Combine like terms: -16.8 + 16.8 = 0.0 16.8 + 7k + 0.2k2 = 0.0 + 27k + -27k 16.8 + 7k + 0.2k2 = 27k + -27k Combine like terms: 27k + -27k = 0 16.8 + 7k + 0.2k2 = 0 Begin completing the square. Divide all terms by 0.2 the coefficient of the squared term: Divide each side by '0.2'. 84 + 35k + k2 = 0 Move the constant term to the right: Add '-84' to each side of the equation. 84 + 35k + -84 + k2 = 0 + -84 Reorder the terms: 84 + -84 + 35k + k2 = 0 + -84 Combine like terms: 84 + -84 = 0 0 + 35k + k2 = 0 + -84 35k + k2 = 0 + -84 Combine like terms: 0 + -84 = -84 35k + k2 = -84 The k term is 35k. Take half its coefficient (17.5). Square it (306.25) and add it to both sides. Add '306.25' to each side of the equation. 35k + 306.25 + k2 = -84 + 306.25 Reorder the terms: 306.25 + 35k + k2 = -84 + 306.25 Combine like terms: -84 + 306.25 = 222.25 306.25 + 35k + k2 = 222.25 Factor a perfect square on the left side: (k + 17.5)(k + 17.5) = 222.25 Calculate the square root of the right side: 14.908051516 Break this problem into two subproblems by setting (k + 17.5) equal to 14.908051516 and -14.908051516.Subproblem 1
k + 17.5 = 14.908051516 Simplifying k + 17.5 = 14.908051516 Reorder the terms: 17.5 + k = 14.908051516 Solving 17.5 + k = 14.908051516 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-17.5' to each side of the equation. 17.5 + -17.5 + k = 14.908051516 + -17.5 Combine like terms: 17.5 + -17.5 = 0.0 0.0 + k = 14.908051516 + -17.5 k = 14.908051516 + -17.5 Combine like terms: 14.908051516 + -17.5 = -2.591948484 k = -2.591948484 Simplifying k = -2.591948484Subproblem 2
k + 17.5 = -14.908051516 Simplifying k + 17.5 = -14.908051516 Reorder the terms: 17.5 + k = -14.908051516 Solving 17.5 + k = -14.908051516 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-17.5' to each side of the equation. 17.5 + -17.5 + k = -14.908051516 + -17.5 Combine like terms: 17.5 + -17.5 = 0.0 0.0 + k = -14.908051516 + -17.5 k = -14.908051516 + -17.5 Combine like terms: -14.908051516 + -17.5 = -32.408051516 k = -32.408051516 Simplifying k = -32.408051516Solution
The solution to the problem is based on the solutions from the subproblems. k = {-2.591948484, -32.408051516}
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