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Simplifying 3n2 + 12n + -57 = -9 Reorder the terms: -57 + 12n + 3n2 = -9 Solving -57 + 12n + 3n2 = -9 Solving for variable 'n'. Reorder the terms: -57 + 9 + 12n + 3n2 = -9 + 9 Combine like terms: -57 + 9 = -48 -48 + 12n + 3n2 = -9 + 9 Combine like terms: -9 + 9 = 0 -48 + 12n + 3n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-16 + 4n + n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-16 + 4n + n2)' equal to zero and attempt to solve: Simplifying -16 + 4n + n2 = 0 Solving -16 + 4n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '16' to each side of the equation. -16 + 4n + 16 + n2 = 0 + 16 Reorder the terms: -16 + 16 + 4n + n2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + 4n + n2 = 0 + 16 4n + n2 = 0 + 16 Combine like terms: 0 + 16 = 16 4n + n2 = 16 The n term is 4n. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4n + 4 + n2 = 16 + 4 Reorder the terms: 4 + 4n + n2 = 16 + 4 Combine like terms: 16 + 4 = 20 4 + 4n + n2 = 20 Factor a perfect square on the left side: (n + 2)(n + 2) = 20 Calculate the square root of the right side: 4.472135955 Break this problem into two subproblems by setting (n + 2) equal to 4.472135955 and -4.472135955.Subproblem 1
n + 2 = 4.472135955 Simplifying n + 2 = 4.472135955 Reorder the terms: 2 + n = 4.472135955 Solving 2 + n = 4.472135955 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 4.472135955 + -2 Combine like terms: 2 + -2 = 0 0 + n = 4.472135955 + -2 n = 4.472135955 + -2 Combine like terms: 4.472135955 + -2 = 2.472135955 n = 2.472135955 Simplifying n = 2.472135955Subproblem 2
n + 2 = -4.472135955 Simplifying n + 2 = -4.472135955 Reorder the terms: 2 + n = -4.472135955 Solving 2 + n = -4.472135955 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -4.472135955 + -2 Combine like terms: 2 + -2 = 0 0 + n = -4.472135955 + -2 n = -4.472135955 + -2 Combine like terms: -4.472135955 + -2 = -6.472135955 n = -6.472135955 Simplifying n = -6.472135955Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.472135955, -6.472135955}Solution
n = {2.472135955, -6.472135955}
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