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Simplifying 3n2 + 9n + 4 = 0 Reorder the terms: 4 + 9n + 3n2 = 0 Solving 4 + 9n + 3n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + 3n + n2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + 3n + -1.333333333 + n2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + 3n + n2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + 3n + n2 = 0 + -1.333333333 3n + n2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 3n + n2 = -1.333333333 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 = -1.333333333 + 2.25 Reorder the terms: 2.25 + 3n + n2 = -1.333333333 + 2.25 Combine like terms: -1.333333333 + 2.25 = 0.916666667 2.25 + 3n + n2 = 0.916666667 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 0.916666667 Calculate the square root of the right side: 0.957427108 Break this problem into two subproblems by setting (n + 1.5) equal to 0.957427108 and -0.957427108.Subproblem 1
n + 1.5 = 0.957427108 Simplifying n + 1.5 = 0.957427108 Reorder the terms: 1.5 + n = 0.957427108 Solving 1.5 + n = 0.957427108 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 = 0.957427108 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 0.957427108 + -1.5 n = 0.957427108 + -1.5 Combine like terms: 0.957427108 + -1.5 = -0.542572892 n = -0.542572892 Simplifying n = -0.542572892Subproblem 2
n + 1.5 = -0.957427108 Simplifying n + 1.5 = -0.957427108 Reorder the terms: 1.5 + n = -0.957427108 Solving 1.5 + n = -0.957427108 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 = -0.957427108 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -0.957427108 + -1.5 n = -0.957427108 + -1.5 Combine like terms: -0.957427108 + -1.5 = -2.457427108 n = -2.457427108 Simplifying n = -2.457427108Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.542572892, -2.457427108}
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