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Simplifying 2k2 + -3 + 12 = k + 60 Reorder the terms: -3 + 12 + 2k2 = k + 60 Combine like terms: -3 + 12 = 9 9 + 2k2 = k + 60 Reorder the terms: 9 + 2k2 = 60 + k Solving 9 + 2k2 = 60 + k Solving for variable 'k'. Reorder the terms: 9 + -60 + -1k + 2k2 = 60 + k + -60 + -1k Combine like terms: 9 + -60 = -51 -51 + -1k + 2k2 = 60 + k + -60 + -1k Reorder the terms: -51 + -1k + 2k2 = 60 + -60 + k + -1k Combine like terms: 60 + -60 = 0 -51 + -1k + 2k2 = 0 + k + -1k -51 + -1k + 2k2 = k + -1k Combine like terms: k + -1k = 0 -51 + -1k + 2k2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -25.5 + -0.5k + k2 = 0 Move the constant term to the right: Add '25.5' to each side of the equation. -25.5 + -0.5k + 25.5 + k2 = 0 + 25.5 Reorder the terms: -25.5 + 25.5 + -0.5k + k2 = 0 + 25.5 Combine like terms: -25.5 + 25.5 = 0.0 0.0 + -0.5k + k2 = 0 + 25.5 -0.5k + k2 = 0 + 25.5 Combine like terms: 0 + 25.5 = 25.5 -0.5k + k2 = 25.5 The k term is -0.5k. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5k + 0.0625 + k2 = 25.5 + 0.0625 Reorder the terms: 0.0625 + -0.5k + k2 = 25.5 + 0.0625 Combine like terms: 25.5 + 0.0625 = 25.5625 0.0625 + -0.5k + k2 = 25.5625 Factor a perfect square on the left side: (k + -0.25)(k + -0.25) = 25.5625 Calculate the square root of the right side: 5.055937104 Break this problem into two subproblems by setting (k + -0.25) equal to 5.055937104 and -5.055937104.Subproblem 1
k + -0.25 = 5.055937104 Simplifying k + -0.25 = 5.055937104 Reorder the terms: -0.25 + k = 5.055937104 Solving -0.25 + k = 5.055937104 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + k = 5.055937104 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + k = 5.055937104 + 0.25 k = 5.055937104 + 0.25 Combine like terms: 5.055937104 + 0.25 = 5.305937104 k = 5.305937104 Simplifying k = 5.305937104Subproblem 2
k + -0.25 = -5.055937104 Simplifying k + -0.25 = -5.055937104 Reorder the terms: -0.25 + k = -5.055937104 Solving -0.25 + k = -5.055937104 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + k = -5.055937104 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + k = -5.055937104 + 0.25 k = -5.055937104 + 0.25 Combine like terms: -5.055937104 + 0.25 = -4.805937104 k = -4.805937104 Simplifying k = -4.805937104Solution
The solution to the problem is based on the solutions from the subproblems. k = {5.305937104, -4.805937104}
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