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Simplifying 12n(n + 3) + 15 = 396 Reorder the terms: 12n(3 + n) + 15 = 396 (3 * 12n + n * 12n) + 15 = 396 (36n + 12n2) + 15 = 396 Reorder the terms: 15 + 36n + 12n2 = 396 Solving 15 + 36n + 12n2 = 396 Solving for variable 'n'. Reorder the terms: 15 + -396 + 36n + 12n2 = 396 + -396 Combine like terms: 15 + -396 = -381 -381 + 36n + 12n2 = 396 + -396 Combine like terms: 396 + -396 = 0 -381 + 36n + 12n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-127 + 12n + 4n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-127 + 12n + 4n2)' equal to zero and attempt to solve: Simplifying -127 + 12n + 4n2 = 0 Solving -127 + 12n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -31.75 + 3n + n2 = 0 Move the constant term to the right: Add '31.75' to each side of the equation. -31.75 + 3n + 31.75 + n2 = 0 + 31.75 Reorder the terms: -31.75 + 31.75 + 3n + n2 = 0 + 31.75 Combine like terms: -31.75 + 31.75 = 0.00 0.00 + 3n + n2 = 0 + 31.75 3n + n2 = 0 + 31.75 Combine like terms: 0 + 31.75 = 31.75 3n + n2 = 31.75 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 31.75 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 31.75 + 2.25 Combine like terms: 31.75 + 2.25 = 34 2.25 + 3n + n2 = 34 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 34 Calculate the square root of the right side: 5.830951895 Break this problem into two subproblems by setting (n + 1.5) equal to 5.830951895 and -5.830951895.Subproblem 1
n + 1.5 = 5.830951895 Simplifying n + 1.5 = 5.830951895 Reorder the terms: 1.5 + n = 5.830951895 Solving 1.5 + n = 5.830951895 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 5.830951895 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 5.830951895 + -1.5 n = 5.830951895 + -1.5 Combine like terms: 5.830951895 + -1.5 = 4.330951895 n = 4.330951895 Simplifying n = 4.330951895Subproblem 2
n + 1.5 = -5.830951895 Simplifying n + 1.5 = -5.830951895 Reorder the terms: 1.5 + n = -5.830951895 Solving 1.5 + n = -5.830951895 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -5.830951895 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -5.830951895 + -1.5 n = -5.830951895 + -1.5 Combine like terms: -5.830951895 + -1.5 = -7.330951895 n = -7.330951895 Simplifying n = -7.330951895Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.330951895, -7.330951895}Solution
n = {4.330951895, -7.330951895}
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