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Simplifying 3n2 + -6n + 2 = 0 Reorder the terms: 2 + -6n + 3n2 = 0 Solving 2 + -6n + 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'. 0.6666666667 + -2n + n2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + -2n + -0.6666666667 + n2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + -2n + n2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + -2n + n2 = 0 + -0.6666666667 -2n + n2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 -2n + n2 = -0.6666666667 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = -0.6666666667 + 1 Reorder the terms: 1 + -2n + n2 = -0.6666666667 + 1 Combine like terms: -0.6666666667 + 1 = 0.3333333333 1 + -2n + n2 = 0.3333333333 Factor a perfect square on the left side: (n + -1)(n + -1) = 0.3333333333 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (n + -1) equal to 0.577350269 and -0.577350269.Subproblem 1
n + -1 = 0.577350269 Simplifying n + -1 = 0.577350269 Reorder the terms: -1 + n = 0.577350269 Solving -1 + n = 0.577350269 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + n = 0.577350269 + 1 n = 0.577350269 + 1 Combine like terms: 0.577350269 + 1 = 1.577350269 n = 1.577350269 Simplifying n = 1.577350269Subproblem 2
n + -1 = -0.577350269 Simplifying n + -1 = -0.577350269 Reorder the terms: -1 + n = -0.577350269 Solving -1 + n = -0.577350269 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + n = -0.577350269 + 1 n = -0.577350269 + 1 Combine like terms: -0.577350269 + 1 = 0.422649731 n = 0.422649731 Simplifying n = 0.422649731Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.577350269, 0.422649731}
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