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Simplifying 3k2 + 12k + 1 = 6 Reorder the terms: 1 + 12k + 3k2 = 6 Solving 1 + 12k + 3k2 = 6 Solving for variable 'k'. Reorder the terms: 1 + -6 + 12k + 3k2 = 6 + -6 Combine like terms: 1 + -6 = -5 -5 + 12k + 3k2 = 6 + -6 Combine like terms: 6 + -6 = 0 -5 + 12k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.666666667 + 4k + k2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 4k + 1.666666667 + k2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 4k + k2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 4k + k2 = 0 + 1.666666667 4k + k2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 4k + k2 = 1.666666667 The k term is 4k. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4k + 4 + k2 = 1.666666667 + 4 Reorder the terms: 4 + 4k + k2 = 1.666666667 + 4 Combine like terms: 1.666666667 + 4 = 5.666666667 4 + 4k + k2 = 5.666666667 Factor a perfect square on the left side: (k + 2)(k + 2) = 5.666666667 Calculate the square root of the right side: 2.380476143 Break this problem into two subproblems by setting (k + 2) equal to 2.380476143 and -2.380476143.Subproblem 1
k + 2 = 2.380476143 Simplifying k + 2 = 2.380476143 Reorder the terms: 2 + k = 2.380476143 Solving 2 + k = 2.380476143 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + k = 2.380476143 + -2 Combine like terms: 2 + -2 = 0 0 + k = 2.380476143 + -2 k = 2.380476143 + -2 Combine like terms: 2.380476143 + -2 = 0.380476143 k = 0.380476143 Simplifying k = 0.380476143Subproblem 2
k + 2 = -2.380476143 Simplifying k + 2 = -2.380476143 Reorder the terms: 2 + k = -2.380476143 Solving 2 + k = -2.380476143 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + k = -2.380476143 + -2 Combine like terms: 2 + -2 = 0 0 + k = -2.380476143 + -2 k = -2.380476143 + -2 Combine like terms: -2.380476143 + -2 = -4.380476143 k = -4.380476143 Simplifying k = -4.380476143Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.380476143, -4.380476143}
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