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Simplifying 2k2 + -3 + 12k = k + 6 Reorder the terms: -3 + 12k + 2k2 = k + 6 Reorder the terms: -3 + 12k + 2k2 = 6 + k Solving -3 + 12k + 2k2 = 6 + k Solving for variable 'k'. Reorder the terms: -3 + -6 + 12k + -1k + 2k2 = 6 + k + -6 + -1k Combine like terms: -3 + -6 = -9 -9 + 12k + -1k + 2k2 = 6 + k + -6 + -1k Combine like terms: 12k + -1k = 11k -9 + 11k + 2k2 = 6 + k + -6 + -1k Reorder the terms: -9 + 11k + 2k2 = 6 + -6 + k + -1k Combine like terms: 6 + -6 = 0 -9 + 11k + 2k2 = 0 + k + -1k -9 + 11k + 2k2 = k + -1k Combine like terms: k + -1k = 0 -9 + 11k + 2k2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4.5 + 5.5k + k2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + 5.5k + 4.5 + k2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + 5.5k + k2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + 5.5k + k2 = 0 + 4.5 5.5k + k2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 5.5k + k2 = 4.5 The k term is 5.5k. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5k + 7.5625 + k2 = 4.5 + 7.5625 Reorder the terms: 7.5625 + 5.5k + k2 = 4.5 + 7.5625 Combine like terms: 4.5 + 7.5625 = 12.0625 7.5625 + 5.5k + k2 = 12.0625 Factor a perfect square on the left side: (k + 2.75)(k + 2.75) = 12.0625 Calculate the square root of the right side: 3.473110997 Break this problem into two subproblems by setting (k + 2.75) equal to 3.473110997 and -3.473110997.Subproblem 1
k + 2.75 = 3.473110997 Simplifying k + 2.75 = 3.473110997 Reorder the terms: 2.75 + k = 3.473110997 Solving 2.75 + k = 3.473110997 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + k = 3.473110997 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + k = 3.473110997 + -2.75 k = 3.473110997 + -2.75 Combine like terms: 3.473110997 + -2.75 = 0.723110997 k = 0.723110997 Simplifying k = 0.723110997Subproblem 2
k + 2.75 = -3.473110997 Simplifying k + 2.75 = -3.473110997 Reorder the terms: 2.75 + k = -3.473110997 Solving 2.75 + k = -3.473110997 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + k = -3.473110997 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + k = -3.473110997 + -2.75 k = -3.473110997 + -2.75 Combine like terms: -3.473110997 + -2.75 = -6.223110997 k = -6.223110997 Simplifying k = -6.223110997Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.723110997, -6.223110997}
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