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Simplifying 0.5(k + 5)(k + 2) = 44 Reorder the terms: 0.5(5 + k)(k + 2) = 44 Reorder the terms: 0.5(5 + k)(2 + k) = 44 Multiply (5 + k) * (2 + k) 0.5(5(2 + k) + k(2 + k)) = 44 0.5((2 * 5 + k * 5) + k(2 + k)) = 44 0.5((10 + 5k) + k(2 + k)) = 44 0.5(10 + 5k + (2 * k + k * k)) = 44 0.5(10 + 5k + (2k + k2)) = 44 Combine like terms: 5k + 2k = 7k 0.5(10 + 7k + k2) = 44 (10 * 0.5 + 7k * 0.5 + k2 * 0.5) = 44 (5 + 3.5k + 0.5k2) = 44 Solving 5 + 3.5k + 0.5k2 = 44 Solving for variable 'k'. Reorder the terms: 5 + -44 + 3.5k + 0.5k2 = 44 + -44 Combine like terms: 5 + -44 = -39 -39 + 3.5k + 0.5k2 = 44 + -44 Combine like terms: 44 + -44 = 0 -39 + 3.5k + 0.5k2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -78 + 7k + k2 = 0 Move the constant term to the right: Add '78' to each side of the equation. -78 + 7k + 78 + k2 = 0 + 78 Reorder the terms: -78 + 78 + 7k + k2 = 0 + 78 Combine like terms: -78 + 78 = 0 0 + 7k + k2 = 0 + 78 7k + k2 = 0 + 78 Combine like terms: 0 + 78 = 78 7k + k2 = 78 The k term is 7k. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7k + 12.25 + k2 = 78 + 12.25 Reorder the terms: 12.25 + 7k + k2 = 78 + 12.25 Combine like terms: 78 + 12.25 = 90.25 12.25 + 7k + k2 = 90.25 Factor a perfect square on the left side: (k + 3.5)(k + 3.5) = 90.25 Calculate the square root of the right side: 9.5 Break this problem into two subproblems by setting (k + 3.5) equal to 9.5 and -9.5.Subproblem 1
k + 3.5 = 9.5 Simplifying k + 3.5 = 9.5 Reorder the terms: 3.5 + k = 9.5 Solving 3.5 + k = 9.5 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + k = 9.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + k = 9.5 + -3.5 k = 9.5 + -3.5 Combine like terms: 9.5 + -3.5 = 6 k = 6 Simplifying k = 6Subproblem 2
k + 3.5 = -9.5 Simplifying k + 3.5 = -9.5 Reorder the terms: 3.5 + k = -9.5 Solving 3.5 + k = -9.5 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + k = -9.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + k = -9.5 + -3.5 k = -9.5 + -3.5 Combine like terms: -9.5 + -3.5 = -13 k = -13 Simplifying k = -13Solution
The solution to the problem is based on the solutions from the subproblems. k = {6, -13}
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