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Simplifying 2k(k + 13) = 13 Reorder the terms: 2k(13 + k) = 13 (13 * 2k + k * 2k) = 13 (26k + 2k2) = 13 Solving 26k + 2k2 = 13 Solving for variable 'k'. Reorder the terms: -13 + 26k + 2k2 = 13 + -13 Combine like terms: 13 + -13 = 0 -13 + 26k + 2k2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6.5 + 13k + k2 = 0 Move the constant term to the right: Add '6.5' to each side of the equation. -6.5 + 13k + 6.5 + k2 = 0 + 6.5 Reorder the terms: -6.5 + 6.5 + 13k + k2 = 0 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + 13k + k2 = 0 + 6.5 13k + k2 = 0 + 6.5 Combine like terms: 0 + 6.5 = 6.5 13k + k2 = 6.5 The k term is 13k. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13k + 42.25 + k2 = 6.5 + 42.25 Reorder the terms: 42.25 + 13k + k2 = 6.5 + 42.25 Combine like terms: 6.5 + 42.25 = 48.75 42.25 + 13k + k2 = 48.75 Factor a perfect square on the left side: (k + 6.5)(k + 6.5) = 48.75 Calculate the square root of the right side: 6.982120022 Break this problem into two subproblems by setting (k + 6.5) equal to 6.982120022 and -6.982120022.Subproblem 1
k + 6.5 = 6.982120022 Simplifying k + 6.5 = 6.982120022 Reorder the terms: 6.5 + k = 6.982120022 Solving 6.5 + k = 6.982120022 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + k = 6.982120022 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + k = 6.982120022 + -6.5 k = 6.982120022 + -6.5 Combine like terms: 6.982120022 + -6.5 = 0.482120022 k = 0.482120022 Simplifying k = 0.482120022Subproblem 2
k + 6.5 = -6.982120022 Simplifying k + 6.5 = -6.982120022 Reorder the terms: 6.5 + k = -6.982120022 Solving 6.5 + k = -6.982120022 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + k = -6.982120022 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + k = -6.982120022 + -6.5 k = -6.982120022 + -6.5 Combine like terms: -6.982120022 + -6.5 = -13.482120022 k = -13.482120022 Simplifying k = -13.482120022Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.482120022, -13.482120022}
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