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Simplifying 2k(k + 11) = 77 Reorder the terms: 2k(11 + k) = 77 (11 * 2k + k * 2k) = 77 (22k + 2k2) = 77 Solving 22k + 2k2 = 77 Solving for variable 'k'. Reorder the terms: -77 + 22k + 2k2 = 77 + -77 Combine like terms: 77 + -77 = 0 -77 + 22k + 2k2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -38.5 + 11k + k2 = 0 Move the constant term to the right: Add '38.5' to each side of the equation. -38.5 + 11k + 38.5 + k2 = 0 + 38.5 Reorder the terms: -38.5 + 38.5 + 11k + k2 = 0 + 38.5 Combine like terms: -38.5 + 38.5 = 0.0 0.0 + 11k + k2 = 0 + 38.5 11k + k2 = 0 + 38.5 Combine like terms: 0 + 38.5 = 38.5 11k + k2 = 38.5 The k term is 11k. Take half its coefficient (5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. 11k + 30.25 + k2 = 38.5 + 30.25 Reorder the terms: 30.25 + 11k + k2 = 38.5 + 30.25 Combine like terms: 38.5 + 30.25 = 68.75 30.25 + 11k + k2 = 68.75 Factor a perfect square on the left side: (k + 5.5)(k + 5.5) = 68.75 Calculate the square root of the right side: 8.291561976 Break this problem into two subproblems by setting (k + 5.5) equal to 8.291561976 and -8.291561976.Subproblem 1
k + 5.5 = 8.291561976 Simplifying k + 5.5 = 8.291561976 Reorder the terms: 5.5 + k = 8.291561976 Solving 5.5 + k = 8.291561976 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + k = 8.291561976 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + k = 8.291561976 + -5.5 k = 8.291561976 + -5.5 Combine like terms: 8.291561976 + -5.5 = 2.791561976 k = 2.791561976 Simplifying k = 2.791561976Subproblem 2
k + 5.5 = -8.291561976 Simplifying k + 5.5 = -8.291561976 Reorder the terms: 5.5 + k = -8.291561976 Solving 5.5 + k = -8.291561976 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + k = -8.291561976 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + k = -8.291561976 + -5.5 k = -8.291561976 + -5.5 Combine like terms: -8.291561976 + -5.5 = -13.791561976 k = -13.791561976 Simplifying k = -13.791561976Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.791561976, -13.791561976}
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