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Simplifying 4n2 = -12n + -4 Reorder the terms: 4n2 = -4 + -12n Solving 4n2 = -4 + -12n Solving for variable 'n'. Reorder the terms: 4 + 12n + 4n2 = -4 + -12n + 4 + 12n Reorder the terms: 4 + 12n + 4n2 = -4 + 4 + -12n + 12n Combine like terms: -4 + 4 = 0 4 + 12n + 4n2 = 0 + -12n + 12n 4 + 12n + 4n2 = -12n + 12n Combine like terms: -12n + 12n = 0 4 + 12n + 4n2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(1 + 3n + n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(1 + 3n + n2)' equal to zero and attempt to solve: Simplifying 1 + 3n + n2 = 0 Solving 1 + 3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 3n + -1 + n2 = 0 + -1 Reorder the terms: 1 + -1 + 3n + n2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 3n + n2 = 0 + -1 3n + n2 = 0 + -1 Combine like terms: 0 + -1 = -1 3n + n2 = -1 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 = -1 + 2.25 Reorder the terms: 2.25 + 3n + n2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + 3n + n2 = 1.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (n + 1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
n + 1.5 = 1.118033989 Simplifying n + 1.5 = 1.118033989 Reorder the terms: 1.5 + n = 1.118033989 Solving 1.5 + n = 1.118033989 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 = 1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 1.118033989 + -1.5 n = 1.118033989 + -1.5 Combine like terms: 1.118033989 + -1.5 = -0.381966011 n = -0.381966011 Simplifying n = -0.381966011Subproblem 2
n + 1.5 = -1.118033989 Simplifying n + 1.5 = -1.118033989 Reorder the terms: 1.5 + n = -1.118033989 Solving 1.5 + n = -1.118033989 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 = -1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -1.118033989 + -1.5 n = -1.118033989 + -1.5 Combine like terms: -1.118033989 + -1.5 = -2.618033989 n = -2.618033989 Simplifying n = -2.618033989Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.381966011, -2.618033989}Solution
n = {-0.381966011, -2.618033989}
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