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Simplifying 3k2 + k = 6 Reorder the terms: k + 3k2 = 6 Solving k + 3k2 = 6 Solving for variable 'k'. Reorder the terms: -6 + k + 3k2 = 6 + -6 Combine like terms: 6 + -6 = 0 -6 + k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2 + 0.3333333333k + k2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.3333333333k + 2 + k2 = 0 + 2 Reorder the terms: -2 + 2 + 0.3333333333k + k2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.3333333333k + k2 = 0 + 2 0.3333333333k + k2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.3333333333k + k2 = 2 The k term is k. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.3333333333k + 0.25 + k2 = 2 + 0.25 Reorder the terms: 0.25 + 0.3333333333k + k2 = 2 + 0.25 Combine like terms: 2 + 0.25 = 2.25 0.25 + 0.3333333333k + k2 = 2.25 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 2.25 Calculate the square root of the right side: 1.5 Break this problem into two subproblems by setting (k + 0.5) equal to 1.5 and -1.5.Subproblem 1
k + 0.5 = 1.5 Simplifying k + 0.5 = 1.5 Reorder the terms: 0.5 + k = 1.5 Solving 0.5 + k = 1.5 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = 1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 1.5 + -0.5 k = 1.5 + -0.5 Combine like terms: 1.5 + -0.5 = 1 k = 1 Simplifying k = 1Subproblem 2
k + 0.5 = -1.5 Simplifying k + 0.5 = -1.5 Reorder the terms: 0.5 + k = -1.5 Solving 0.5 + k = -1.5 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = -1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -1.5 + -0.5 k = -1.5 + -0.5 Combine like terms: -1.5 + -0.5 = -2 k = -2 Simplifying k = -2Solution
The solution to the problem is based on the solutions from the subproblems. k = {1, -2}
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