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Simplifying 12k2 + -36k + 27 = k Reorder the terms: 27 + -36k + 12k2 = k Solving 27 + -36k + 12k2 = k Solving for variable 'k'. Reorder the terms: 27 + -36k + -1k + 12k2 = k + -1k Combine like terms: -36k + -1k = -37k 27 + -37k + 12k2 = k + -1k Combine like terms: k + -1k = 0 27 + -37k + 12k2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 2.25 + -3.083333333k + k2 = 0 Move the constant term to the right: Add '-2.25' to each side of the equation. 2.25 + -3.083333333k + -2.25 + k2 = 0 + -2.25 Reorder the terms: 2.25 + -2.25 + -3.083333333k + k2 = 0 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + -3.083333333k + k2 = 0 + -2.25 -3.083333333k + k2 = 0 + -2.25 Combine like terms: 0 + -2.25 = -2.25 -3.083333333k + k2 = -2.25 The k term is -3.083333333k. Take half its coefficient (-1.541666667). Square it (2.376736112) and add it to both sides. Add '2.376736112' to each side of the equation. -3.083333333k + 2.376736112 + k2 = -2.25 + 2.376736112 Reorder the terms: 2.376736112 + -3.083333333k + k2 = -2.25 + 2.376736112 Combine like terms: -2.25 + 2.376736112 = 0.126736112 2.376736112 + -3.083333333k + k2 = 0.126736112 Factor a perfect square on the left side: (k + -1.541666667)(k + -1.541666667) = 0.126736112 Calculate the square root of the right side: 0.356000157 Break this problem into two subproblems by setting (k + -1.541666667) equal to 0.356000157 and -0.356000157.Subproblem 1
k + -1.541666667 = 0.356000157 Simplifying k + -1.541666667 = 0.356000157 Reorder the terms: -1.541666667 + k = 0.356000157 Solving -1.541666667 + k = 0.356000157 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.541666667' to each side of the equation. -1.541666667 + 1.541666667 + k = 0.356000157 + 1.541666667 Combine like terms: -1.541666667 + 1.541666667 = 0.000000000 0.000000000 + k = 0.356000157 + 1.541666667 k = 0.356000157 + 1.541666667 Combine like terms: 0.356000157 + 1.541666667 = 1.897666824 k = 1.897666824 Simplifying k = 1.897666824Subproblem 2
k + -1.541666667 = -0.356000157 Simplifying k + -1.541666667 = -0.356000157 Reorder the terms: -1.541666667 + k = -0.356000157 Solving -1.541666667 + k = -0.356000157 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.541666667' to each side of the equation. -1.541666667 + 1.541666667 + k = -0.356000157 + 1.541666667 Combine like terms: -1.541666667 + 1.541666667 = 0.000000000 0.000000000 + k = -0.356000157 + 1.541666667 k = -0.356000157 + 1.541666667 Combine like terms: -0.356000157 + 1.541666667 = 1.18566651 k = 1.18566651 Simplifying k = 1.18566651Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.897666824, 1.18566651}
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