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Simplifying (2n + -1)(n + 5) = 9 Reorder the terms: (-1 + 2n)(n + 5) = 9 Reorder the terms: (-1 + 2n)(5 + n) = 9 Multiply (-1 + 2n) * (5 + n) (-1(5 + n) + 2n * (5 + n)) = 9 ((5 * -1 + n * -1) + 2n * (5 + n)) = 9 ((-5 + -1n) + 2n * (5 + n)) = 9 (-5 + -1n + (5 * 2n + n * 2n)) = 9 (-5 + -1n + (10n + 2n2)) = 9 Combine like terms: -1n + 10n = 9n (-5 + 9n + 2n2) = 9 Solving -5 + 9n + 2n2 = 9 Solving for variable 'n'. Reorder the terms: -5 + -9 + 9n + 2n2 = 9 + -9 Combine like terms: -5 + -9 = -14 -14 + 9n + 2n2 = 9 + -9 Combine like terms: 9 + -9 = 0 -14 + 9n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7 + 4.5n + n2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + 4.5n + 7 + n2 = 0 + 7 Reorder the terms: -7 + 7 + 4.5n + n2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 4.5n + n2 = 0 + 7 4.5n + n2 = 0 + 7 Combine like terms: 0 + 7 = 7 4.5n + n2 = 7 The n term is 4.5n. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5n + 5.0625 + n2 = 7 + 5.0625 Reorder the terms: 5.0625 + 4.5n + n2 = 7 + 5.0625 Combine like terms: 7 + 5.0625 = 12.0625 5.0625 + 4.5n + n2 = 12.0625 Factor a perfect square on the left side: (n + 2.25)(n + 2.25) = 12.0625 Calculate the square root of the right side: 3.473110997 Break this problem into two subproblems by setting (n + 2.25) equal to 3.473110997 and -3.473110997.Subproblem 1
n + 2.25 = 3.473110997 Simplifying n + 2.25 = 3.473110997 Reorder the terms: 2.25 + n = 3.473110997 Solving 2.25 + n = 3.473110997 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + n = 3.473110997 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + n = 3.473110997 + -2.25 n = 3.473110997 + -2.25 Combine like terms: 3.473110997 + -2.25 = 1.223110997 n = 1.223110997 Simplifying n = 1.223110997Subproblem 2
n + 2.25 = -3.473110997 Simplifying n + 2.25 = -3.473110997 Reorder the terms: 2.25 + n = -3.473110997 Solving 2.25 + n = -3.473110997 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + n = -3.473110997 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + n = -3.473110997 + -2.25 n = -3.473110997 + -2.25 Combine like terms: -3.473110997 + -2.25 = -5.723110997 n = -5.723110997 Simplifying n = -5.723110997Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.223110997, -5.723110997}
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