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Simplifying (3n + 5)(8n + -9) = 180 Reorder the terms: (5 + 3n)(8n + -9) = 180 Reorder the terms: (5 + 3n)(-9 + 8n) = 180 Multiply (5 + 3n) * (-9 + 8n) (5(-9 + 8n) + 3n * (-9 + 8n)) = 180 ((-9 * 5 + 8n * 5) + 3n * (-9 + 8n)) = 180 ((-45 + 40n) + 3n * (-9 + 8n)) = 180 (-45 + 40n + (-9 * 3n + 8n * 3n)) = 180 (-45 + 40n + (-27n + 24n2)) = 180 Combine like terms: 40n + -27n = 13n (-45 + 13n + 24n2) = 180 Solving -45 + 13n + 24n2 = 180 Solving for variable 'n'. Reorder the terms: -45 + -180 + 13n + 24n2 = 180 + -180 Combine like terms: -45 + -180 = -225 -225 + 13n + 24n2 = 180 + -180 Combine like terms: 180 + -180 = 0 -225 + 13n + 24n2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -9.375 + 0.5416666667n + n2 = 0 Move the constant term to the right: Add '9.375' to each side of the equation. -9.375 + 0.5416666667n + 9.375 + n2 = 0 + 9.375 Reorder the terms: -9.375 + 9.375 + 0.5416666667n + n2 = 0 + 9.375 Combine like terms: -9.375 + 9.375 = 0.000 0.000 + 0.5416666667n + n2 = 0 + 9.375 0.5416666667n + n2 = 0 + 9.375 Combine like terms: 0 + 9.375 = 9.375 0.5416666667n + n2 = 9.375 The n term is 0.5416666667n. Take half its coefficient (0.2708333334). Square it (0.07335069448) and add it to both sides. Add '0.07335069448' to each side of the equation. 0.5416666667n + 0.07335069448 + n2 = 9.375 + 0.07335069448 Reorder the terms: 0.07335069448 + 0.5416666667n + n2 = 9.375 + 0.07335069448 Combine like terms: 9.375 + 0.07335069448 = 9.44835069448 0.07335069448 + 0.5416666667n + n2 = 9.44835069448 Factor a perfect square on the left side: (n + 0.2708333334)(n + 0.2708333334) = 9.44835069448 Calculate the square root of the right side: 3.073816959 Break this problem into two subproblems by setting (n + 0.2708333334) equal to 3.073816959 and -3.073816959.Subproblem 1
n + 0.2708333334 = 3.073816959 Simplifying n + 0.2708333334 = 3.073816959 Reorder the terms: 0.2708333334 + n = 3.073816959 Solving 0.2708333334 + n = 3.073816959 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2708333334' to each side of the equation. 0.2708333334 + -0.2708333334 + n = 3.073816959 + -0.2708333334 Combine like terms: 0.2708333334 + -0.2708333334 = 0.0000000000 0.0000000000 + n = 3.073816959 + -0.2708333334 n = 3.073816959 + -0.2708333334 Combine like terms: 3.073816959 + -0.2708333334 = 2.8029836256 n = 2.8029836256 Simplifying n = 2.8029836256Subproblem 2
n + 0.2708333334 = -3.073816959 Simplifying n + 0.2708333334 = -3.073816959 Reorder the terms: 0.2708333334 + n = -3.073816959 Solving 0.2708333334 + n = -3.073816959 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2708333334' to each side of the equation. 0.2708333334 + -0.2708333334 + n = -3.073816959 + -0.2708333334 Combine like terms: 0.2708333334 + -0.2708333334 = 0.0000000000 0.0000000000 + n = -3.073816959 + -0.2708333334 n = -3.073816959 + -0.2708333334 Combine like terms: -3.073816959 + -0.2708333334 = -3.3446502924 n = -3.3446502924 Simplifying n = -3.3446502924Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.8029836256, -3.3446502924}
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