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Simplifying 0 = 3n2 + 3n + -440 Reorder the terms: 0 = -440 + 3n + 3n2 Solving 0 = -440 + 3n + 3n2 Solving for variable 'n'. Combine like terms: 0 + 440 = 440 440 + -3n + -3n2 = -440 + 3n + 3n2 + 440 + -3n + -3n2 Reorder the terms: 440 + -3n + -3n2 = -440 + 440 + 3n + -3n + 3n2 + -3n2 Combine like terms: -440 + 440 = 0 440 + -3n + -3n2 = 0 + 3n + -3n + 3n2 + -3n2 440 + -3n + -3n2 = 3n + -3n + 3n2 + -3n2 Combine like terms: 3n + -3n = 0 440 + -3n + -3n2 = 0 + 3n2 + -3n2 440 + -3n + -3n2 = 3n2 + -3n2 Combine like terms: 3n2 + -3n2 = 0 440 + -3n + -3n2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -146.6666667 + n + n2 = 0 Move the constant term to the right: Add '146.6666667' to each side of the equation. -146.6666667 + n + 146.6666667 + n2 = 0 + 146.6666667 Reorder the terms: -146.6666667 + 146.6666667 + n + n2 = 0 + 146.6666667 Combine like terms: -146.6666667 + 146.6666667 = 0.0000000 0.0000000 + n + n2 = 0 + 146.6666667 n + n2 = 0 + 146.6666667 Combine like terms: 0 + 146.6666667 = 146.6666667 n + n2 = 146.6666667 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + n2 = 146.6666667 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = 146.6666667 + 0.25 Combine like terms: 146.6666667 + 0.25 = 146.9166667 1.25 + n2 = 146.9166667 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 146.9166667 Calculate the square root of the right side: 12.120918558 Break this problem into two subproblems by setting (n + 0.5) equal to 12.120918558 and -12.120918558.Subproblem 1
n + 0.5 = 12.120918558 Simplifying n + 0.5 = 12.120918558 Reorder the terms: 0.5 + n = 12.120918558 Solving 0.5 + n = 12.120918558 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 12.120918558 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 12.120918558 + -0.5 n = 12.120918558 + -0.5 Combine like terms: 12.120918558 + -0.5 = 11.620918558 n = 11.620918558 Simplifying n = 11.620918558Subproblem 2
n + 0.5 = -12.120918558 Simplifying n + 0.5 = -12.120918558 Reorder the terms: 0.5 + n = -12.120918558 Solving 0.5 + n = -12.120918558 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -12.120918558 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -12.120918558 + -0.5 n = -12.120918558 + -0.5 Combine like terms: -12.120918558 + -0.5 = -12.620918558 n = -12.620918558 Simplifying n = -12.620918558Solution
The solution to the problem is based on the solutions from the subproblems. n = {11.620918558, -12.620918558}
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