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Simplifying 3n2 + 12n + -27 = 0 Reorder the terms: -27 + 12n + 3n2 = 0 Solving -27 + 12n + 3n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '3'. 3(-9 + 4n + n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-9 + 4n + n2)' equal to zero and attempt to solve: Simplifying -9 + 4n + n2 = 0 Solving -9 + 4n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '9' to each side of the equation. -9 + 4n + 9 + n2 = 0 + 9 Reorder the terms: -9 + 9 + 4n + n2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 4n + n2 = 0 + 9 4n + n2 = 0 + 9 Combine like terms: 0 + 9 = 9 4n + n2 = 9 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 = 9 + 4 Reorder the terms: 4 + 4n + n2 = 9 + 4 Combine like terms: 9 + 4 = 13 4 + 4n + n2 = 13 Factor a perfect square on the left side: (n + 2)(n + 2) = 13 Calculate the square root of the right side: 3.605551275 Break this problem into two subproblems by setting (n + 2) equal to 3.605551275 and -3.605551275.Subproblem 1
n + 2 = 3.605551275 Simplifying n + 2 = 3.605551275 Reorder the terms: 2 + n = 3.605551275 Solving 2 + n = 3.605551275 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 = 3.605551275 + -2 Combine like terms: 2 + -2 = 0 0 + n = 3.605551275 + -2 n = 3.605551275 + -2 Combine like terms: 3.605551275 + -2 = 1.605551275 n = 1.605551275 Simplifying n = 1.605551275Subproblem 2
n + 2 = -3.605551275 Simplifying n + 2 = -3.605551275 Reorder the terms: 2 + n = -3.605551275 Solving 2 + n = -3.605551275 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 = -3.605551275 + -2 Combine like terms: 2 + -2 = 0 0 + n = -3.605551275 + -2 n = -3.605551275 + -2 Combine like terms: -3.605551275 + -2 = -5.605551275 n = -5.605551275 Simplifying n = -5.605551275Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.605551275, -5.605551275}Solution
n = {1.605551275, -5.605551275}
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