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Simplifying 3n2 + 108 = 37n Reorder the terms: 108 + 3n2 = 37n Solving 108 + 3n2 = 37n Solving for variable 'n'. Reorder the terms: 108 + -37n + 3n2 = 37n + -37n Combine like terms: 37n + -37n = 0 108 + -37n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 36 + -12.33333333n + n2 = 0 Move the constant term to the right: Add '-36' to each side of the equation. 36 + -12.33333333n + -36 + n2 = 0 + -36 Reorder the terms: 36 + -36 + -12.33333333n + n2 = 0 + -36 Combine like terms: 36 + -36 = 0 0 + -12.33333333n + n2 = 0 + -36 -12.33333333n + n2 = 0 + -36 Combine like terms: 0 + -36 = -36 -12.33333333n + n2 = -36 The n term is -12.33333333n. Take half its coefficient (-6.166666665). Square it (38.02777776) and add it to both sides. Add '38.02777776' to each side of the equation. -12.33333333n + 38.02777776 + n2 = -36 + 38.02777776 Reorder the terms: 38.02777776 + -12.33333333n + n2 = -36 + 38.02777776 Combine like terms: -36 + 38.02777776 = 2.02777776 38.02777776 + -12.33333333n + n2 = 2.02777776 Factor a perfect square on the left side: (n + -6.166666665)(n + -6.166666665) = 2.02777776 Calculate the square root of the right side: 1.424000618 Break this problem into two subproblems by setting (n + -6.166666665) equal to 1.424000618 and -1.424000618.Subproblem 1
n + -6.166666665 = 1.424000618 Simplifying n + -6.166666665 = 1.424000618 Reorder the terms: -6.166666665 + n = 1.424000618 Solving -6.166666665 + n = 1.424000618 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.166666665' to each side of the equation. -6.166666665 + 6.166666665 + n = 1.424000618 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + n = 1.424000618 + 6.166666665 n = 1.424000618 + 6.166666665 Combine like terms: 1.424000618 + 6.166666665 = 7.590667283 n = 7.590667283 Simplifying n = 7.590667283Subproblem 2
n + -6.166666665 = -1.424000618 Simplifying n + -6.166666665 = -1.424000618 Reorder the terms: -6.166666665 + n = -1.424000618 Solving -6.166666665 + n = -1.424000618 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.166666665' to each side of the equation. -6.166666665 + 6.166666665 + n = -1.424000618 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + n = -1.424000618 + 6.166666665 n = -1.424000618 + 6.166666665 Combine like terms: -1.424000618 + 6.166666665 = 4.742666047 n = 4.742666047 Simplifying n = 4.742666047Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.590667283, 4.742666047}
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