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Simplifying -4k2 + -9k + -3k = -3 + -5k2 Reorder the terms: -9k + -3k + -4k2 = -3 + -5k2 Combine like terms: -9k + -3k = -12k -12k + -4k2 = -3 + -5k2 Solving -12k + -4k2 = -3 + -5k2 Solving for variable 'k'. Reorder the terms: 3 + -12k + -4k2 + 5k2 = -3 + -5k2 + 3 + 5k2 Combine like terms: -4k2 + 5k2 = 1k2 3 + -12k + 1k2 = -3 + -5k2 + 3 + 5k2 Reorder the terms: 3 + -12k + 1k2 = -3 + 3 + -5k2 + 5k2 Combine like terms: -3 + 3 = 0 3 + -12k + 1k2 = 0 + -5k2 + 5k2 3 + -12k + 1k2 = -5k2 + 5k2 Combine like terms: -5k2 + 5k2 = 0 3 + -12k + 1k2 = 0 Begin completing the square. Move the constant term to the right: Add '-3' to each side of the equation. 3 + -12k + -3 + k2 = 0 + -3 Reorder the terms: 3 + -3 + -12k + k2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -12k + k2 = 0 + -3 -12k + k2 = 0 + -3 Combine like terms: 0 + -3 = -3 -12k + k2 = -3 The k term is -12k. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12k + 36 + k2 = -3 + 36 Reorder the terms: 36 + -12k + k2 = -3 + 36 Combine like terms: -3 + 36 = 33 36 + -12k + k2 = 33 Factor a perfect square on the left side: (k + -6)(k + -6) = 33 Calculate the square root of the right side: 5.744562647 Break this problem into two subproblems by setting (k + -6) equal to 5.744562647 and -5.744562647.Subproblem 1
k + -6 = 5.744562647 Simplifying k + -6 = 5.744562647 Reorder the terms: -6 + k = 5.744562647 Solving -6 + k = 5.744562647 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + k = 5.744562647 + 6 Combine like terms: -6 + 6 = 0 0 + k = 5.744562647 + 6 k = 5.744562647 + 6 Combine like terms: 5.744562647 + 6 = 11.744562647 k = 11.744562647 Simplifying k = 11.744562647Subproblem 2
k + -6 = -5.744562647 Simplifying k + -6 = -5.744562647 Reorder the terms: -6 + k = -5.744562647 Solving -6 + k = -5.744562647 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + k = -5.744562647 + 6 Combine like terms: -6 + 6 = 0 0 + k = -5.744562647 + 6 k = -5.744562647 + 6 Combine like terms: -5.744562647 + 6 = 0.255437353 k = 0.255437353 Simplifying k = 0.255437353Solution
The solution to the problem is based on the solutions from the subproblems. k = {11.744562647, 0.255437353}
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