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Simplifying 9k2 + -22k + -25 = 0 Reorder the terms: -25 + -22k + 9k2 = 0 Solving -25 + -22k + 9k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.777777778 + -2.444444444k + k2 = 0 Move the constant term to the right: Add '2.777777778' to each side of the equation. -2.777777778 + -2.444444444k + 2.777777778 + k2 = 0 + 2.777777778 Reorder the terms: -2.777777778 + 2.777777778 + -2.444444444k + k2 = 0 + 2.777777778 Combine like terms: -2.777777778 + 2.777777778 = 0.000000000 0.000000000 + -2.444444444k + k2 = 0 + 2.777777778 -2.444444444k + k2 = 0 + 2.777777778 Combine like terms: 0 + 2.777777778 = 2.777777778 -2.444444444k + k2 = 2.777777778 The k term is -2.444444444k. Take half its coefficient (-1.222222222). Square it (1.493827160) and add it to both sides. Add '1.493827160' to each side of the equation. -2.444444444k + 1.493827160 + k2 = 2.777777778 + 1.493827160 Reorder the terms: 1.493827160 + -2.444444444k + k2 = 2.777777778 + 1.493827160 Combine like terms: 2.777777778 + 1.493827160 = 4.271604938 1.493827160 + -2.444444444k + k2 = 4.271604938 Factor a perfect square on the left side: (k + -1.222222222)(k + -1.222222222) = 4.271604938 Calculate the square root of the right side: 2.066786137 Break this problem into two subproblems by setting (k + -1.222222222) equal to 2.066786137 and -2.066786137.Subproblem 1
k + -1.222222222 = 2.066786137 Simplifying k + -1.222222222 = 2.066786137 Reorder the terms: -1.222222222 + k = 2.066786137 Solving -1.222222222 + k = 2.066786137 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.222222222' to each side of the equation. -1.222222222 + 1.222222222 + k = 2.066786137 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + k = 2.066786137 + 1.222222222 k = 2.066786137 + 1.222222222 Combine like terms: 2.066786137 + 1.222222222 = 3.289008359 k = 3.289008359 Simplifying k = 3.289008359Subproblem 2
k + -1.222222222 = -2.066786137 Simplifying k + -1.222222222 = -2.066786137 Reorder the terms: -1.222222222 + k = -2.066786137 Solving -1.222222222 + k = -2.066786137 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.222222222' to each side of the equation. -1.222222222 + 1.222222222 + k = -2.066786137 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + k = -2.066786137 + 1.222222222 k = -2.066786137 + 1.222222222 Combine like terms: -2.066786137 + 1.222222222 = -0.844563915 k = -0.844563915 Simplifying k = -0.844563915Solution
The solution to the problem is based on the solutions from the subproblems. k = {3.289008359, -0.844563915}
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