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Simplifying 3n2 = 6 + -12n Solving 3n2 = 6 + -12n Solving for variable 'n'. Reorder the terms: -6 + 12n + 3n2 = 6 + -12n + -6 + 12n Reorder the terms: -6 + 12n + 3n2 = 6 + -6 + -12n + 12n Combine like terms: 6 + -6 = 0 -6 + 12n + 3n2 = 0 + -12n + 12n -6 + 12n + 3n2 = -12n + 12n Combine like terms: -12n + 12n = 0 -6 + 12n + 3n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-2 + 4n + n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-2 + 4n + n2)' equal to zero and attempt to solve: Simplifying -2 + 4n + n2 = 0 Solving -2 + 4n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '2' to each side of the equation. -2 + 4n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + 4n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 4n + n2 = 0 + 2 4n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 4n + n2 = 2 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 = 2 + 4 Reorder the terms: 4 + 4n + n2 = 2 + 4 Combine like terms: 2 + 4 = 6 4 + 4n + n2 = 6 Factor a perfect square on the left side: (n + 2)(n + 2) = 6 Calculate the square root of the right side: 2.449489743 Break this problem into two subproblems by setting (n + 2) equal to 2.449489743 and -2.449489743.Subproblem 1
n + 2 = 2.449489743 Simplifying n + 2 = 2.449489743 Reorder the terms: 2 + n = 2.449489743 Solving 2 + n = 2.449489743 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 = 2.449489743 + -2 Combine like terms: 2 + -2 = 0 0 + n = 2.449489743 + -2 n = 2.449489743 + -2 Combine like terms: 2.449489743 + -2 = 0.449489743 n = 0.449489743 Simplifying n = 0.449489743Subproblem 2
n + 2 = -2.449489743 Simplifying n + 2 = -2.449489743 Reorder the terms: 2 + n = -2.449489743 Solving 2 + n = -2.449489743 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 = -2.449489743 + -2 Combine like terms: 2 + -2 = 0 0 + n = -2.449489743 + -2 n = -2.449489743 + -2 Combine like terms: -2.449489743 + -2 = -4.449489743 n = -4.449489743 Simplifying n = -4.449489743Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.449489743, -4.449489743}Solution
n = {0.449489743, -4.449489743}
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