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Simplifying 0.6 = -6.1k + 1.3k(2k + 2) Reorder the terms: 0.6 = -6.1k + 1.3k(2 + 2k) 0.6 = -6.1k + (2 * 1.3k + 2k * 1.3k) 0.6 = -6.1k + (2.6k + 2.6k2) Combine like terms: -6.1k + 2.6k = -3.5k 0.6 = -3.5k + 2.6k2 Solving 0.6 = -3.5k + 2.6k2 Solving for variable 'k'. Reorder the terms: 0.6 + 3.5k + -2.6k2 = -3.5k + 3.5k + 2.6k2 + -2.6k2 Combine like terms: -3.5k + 3.5k = 0.0 0.6 + 3.5k + -2.6k2 = 0.0 + 2.6k2 + -2.6k2 0.6 + 3.5k + -2.6k2 = 2.6k2 + -2.6k2 Combine like terms: 2.6k2 + -2.6k2 = 0.0 0.6 + 3.5k + -2.6k2 = 0.0 Begin completing the square. Divide all terms by -2.6 the coefficient of the squared term: Divide each side by '-2.6'. -0.2307692308 + -1.346153846k + k2 = 0 Move the constant term to the right: Add '0.2307692308' to each side of the equation. -0.2307692308 + -1.346153846k + 0.2307692308 + k2 = 0 + 0.2307692308 Reorder the terms: -0.2307692308 + 0.2307692308 + -1.346153846k + k2 = 0 + 0.2307692308 Combine like terms: -0.2307692308 + 0.2307692308 = 0.0000000000 0.0000000000 + -1.346153846k + k2 = 0 + 0.2307692308 -1.346153846k + k2 = 0 + 0.2307692308 Combine like terms: 0 + 0.2307692308 = 0.2307692308 -1.346153846k + k2 = 0.2307692308 The k term is -1.346153846k. Take half its coefficient (-0.673076923). Square it (0.4530325443) and add it to both sides. Add '0.4530325443' to each side of the equation. -1.346153846k + 0.4530325443 + k2 = 0.2307692308 + 0.4530325443 Reorder the terms: 0.4530325443 + -1.346153846k + k2 = 0.2307692308 + 0.4530325443 Combine like terms: 0.2307692308 + 0.4530325443 = 0.6838017751 0.4530325443 + -1.346153846k + k2 = 0.6838017751 Factor a perfect square on the left side: (k + -0.673076923)(k + -0.673076923) = 0.6838017751 Calculate the square root of the right side: 0.826923077 Break this problem into two subproblems by setting (k + -0.673076923) equal to 0.826923077 and -0.826923077.Subproblem 1
k + -0.673076923 = 0.826923077 Simplifying k + -0.673076923 = 0.826923077 Reorder the terms: -0.673076923 + k = 0.826923077 Solving -0.673076923 + k = 0.826923077 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.673076923' to each side of the equation. -0.673076923 + 0.673076923 + k = 0.826923077 + 0.673076923 Combine like terms: -0.673076923 + 0.673076923 = 0.000000000 0.000000000 + k = 0.826923077 + 0.673076923 k = 0.826923077 + 0.673076923 Combine like terms: 0.826923077 + 0.673076923 = 1.5 k = 1.5 Simplifying k = 1.5Subproblem 2
k + -0.673076923 = -0.826923077 Simplifying k + -0.673076923 = -0.826923077 Reorder the terms: -0.673076923 + k = -0.826923077 Solving -0.673076923 + k = -0.826923077 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.673076923' to each side of the equation. -0.673076923 + 0.673076923 + k = -0.826923077 + 0.673076923 Combine like terms: -0.673076923 + 0.673076923 = 0.000000000 0.000000000 + k = -0.826923077 + 0.673076923 k = -0.826923077 + 0.673076923 Combine like terms: -0.826923077 + 0.673076923 = -0.153846154 k = -0.153846154 Simplifying k = -0.153846154Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.5, -0.153846154}
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