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Simplifying -11k2 + -34k + 1 = 0 Reorder the terms: 1 + -34k + -11k2 = 0 Solving 1 + -34k + -11k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by -11 the coefficient of the squared term: Divide each side by '-11'. -0.09090909091 + 3.090909091k + k2 = 0 Move the constant term to the right: Add '0.09090909091' to each side of the equation. -0.09090909091 + 3.090909091k + 0.09090909091 + k2 = 0 + 0.09090909091 Reorder the terms: -0.09090909091 + 0.09090909091 + 3.090909091k + k2 = 0 + 0.09090909091 Combine like terms: -0.09090909091 + 0.09090909091 = 0.00000000000 0.00000000000 + 3.090909091k + k2 = 0 + 0.09090909091 3.090909091k + k2 = 0 + 0.09090909091 Combine like terms: 0 + 0.09090909091 = 0.09090909091 3.090909091k + k2 = 0.09090909091 The k term is 3.090909091k. Take half its coefficient (1.545454546). Square it (2.388429754) and add it to both sides. Add '2.388429754' to each side of the equation. 3.090909091k + 2.388429754 + k2 = 0.09090909091 + 2.388429754 Reorder the terms: 2.388429754 + 3.090909091k + k2 = 0.09090909091 + 2.388429754 Combine like terms: 0.09090909091 + 2.388429754 = 2.47933884491 2.388429754 + 3.090909091k + k2 = 2.47933884491 Factor a perfect square on the left side: (k + 1.545454546)(k + 1.545454546) = 2.47933884491 Calculate the square root of the right side: 1.574591644 Break this problem into two subproblems by setting (k + 1.545454546) equal to 1.574591644 and -1.574591644.Subproblem 1
k + 1.545454546 = 1.574591644 Simplifying k + 1.545454546 = 1.574591644 Reorder the terms: 1.545454546 + k = 1.574591644 Solving 1.545454546 + k = 1.574591644 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.545454546' to each side of the equation. 1.545454546 + -1.545454546 + k = 1.574591644 + -1.545454546 Combine like terms: 1.545454546 + -1.545454546 = 0.000000000 0.000000000 + k = 1.574591644 + -1.545454546 k = 1.574591644 + -1.545454546 Combine like terms: 1.574591644 + -1.545454546 = 0.029137098 k = 0.029137098 Simplifying k = 0.029137098Subproblem 2
k + 1.545454546 = -1.574591644 Simplifying k + 1.545454546 = -1.574591644 Reorder the terms: 1.545454546 + k = -1.574591644 Solving 1.545454546 + k = -1.574591644 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.545454546' to each side of the equation. 1.545454546 + -1.545454546 + k = -1.574591644 + -1.545454546 Combine like terms: 1.545454546 + -1.545454546 = 0.000000000 0.000000000 + k = -1.574591644 + -1.545454546 k = -1.574591644 + -1.545454546 Combine like terms: -1.574591644 + -1.545454546 = -3.12004619 k = -3.12004619 Simplifying k = -3.12004619Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.029137098, -3.12004619}
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