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Simplifying 3n2 + 12n = 12 Reorder the terms: 12n + 3n2 = 12 Solving 12n + 3n2 = 12 Solving for variable 'n'. Reorder the terms: -12 + 12n + 3n2 = 12 + -12 Combine like terms: 12 + -12 = 0 -12 + 12n + 3n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-4 + 4n + n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-4 + 4n + n2)' equal to zero and attempt to solve: Simplifying -4 + 4n + n2 = 0 Solving -4 + 4n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + 4n + 4 + n2 = 0 + 4 Reorder the terms: -4 + 4 + 4n + n2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 4n + n2 = 0 + 4 4n + n2 = 0 + 4 Combine like terms: 0 + 4 = 4 4n + n2 = 4 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 = 4 + 4 Reorder the terms: 4 + 4n + n2 = 4 + 4 Combine like terms: 4 + 4 = 8 4 + 4n + n2 = 8 Factor a perfect square on the left side: (n + 2)(n + 2) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (n + 2) equal to 2.828427125 and -2.828427125.Subproblem 1
n + 2 = 2.828427125 Simplifying n + 2 = 2.828427125 Reorder the terms: 2 + n = 2.828427125 Solving 2 + n = 2.828427125 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 = 2.828427125 + -2 Combine like terms: 2 + -2 = 0 0 + n = 2.828427125 + -2 n = 2.828427125 + -2 Combine like terms: 2.828427125 + -2 = 0.828427125 n = 0.828427125 Simplifying n = 0.828427125Subproblem 2
n + 2 = -2.828427125 Simplifying n + 2 = -2.828427125 Reorder the terms: 2 + n = -2.828427125 Solving 2 + n = -2.828427125 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 = -2.828427125 + -2 Combine like terms: 2 + -2 = 0 0 + n = -2.828427125 + -2 n = -2.828427125 + -2 Combine like terms: -2.828427125 + -2 = -4.828427125 n = -4.828427125 Simplifying n = -4.828427125Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.828427125, -4.828427125}Solution
n = {0.828427125, -4.828427125}
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