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Simplifying 3n2 + 21n + -34 = 0 Reorder the terms: -34 + 21n + 3n2 = 0 Solving -34 + 21n + 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'. -11.33333333 + 7n + n2 = 0 Move the constant term to the right: Add '11.33333333' to each side of the equation. -11.33333333 + 7n + 11.33333333 + n2 = 0 + 11.33333333 Reorder the terms: -11.33333333 + 11.33333333 + 7n + n2 = 0 + 11.33333333 Combine like terms: -11.33333333 + 11.33333333 = 0.00000000 0.00000000 + 7n + n2 = 0 + 11.33333333 7n + n2 = 0 + 11.33333333 Combine like terms: 0 + 11.33333333 = 11.33333333 7n + n2 = 11.33333333 The n term is 7n. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7n + 12.25 + n2 = 11.33333333 + 12.25 Reorder the terms: 12.25 + 7n + n2 = 11.33333333 + 12.25 Combine like terms: 11.33333333 + 12.25 = 23.58333333 12.25 + 7n + n2 = 23.58333333 Factor a perfect square on the left side: (n + 3.5)(n + 3.5) = 23.58333333 Calculate the square root of the right side: 4.856267428 Break this problem into two subproblems by setting (n + 3.5) equal to 4.856267428 and -4.856267428.Subproblem 1
n + 3.5 = 4.856267428 Simplifying n + 3.5 = 4.856267428 Reorder the terms: 3.5 + n = 4.856267428 Solving 3.5 + n = 4.856267428 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = 4.856267428 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 4.856267428 + -3.5 n = 4.856267428 + -3.5 Combine like terms: 4.856267428 + -3.5 = 1.356267428 n = 1.356267428 Simplifying n = 1.356267428Subproblem 2
n + 3.5 = -4.856267428 Simplifying n + 3.5 = -4.856267428 Reorder the terms: 3.5 + n = -4.856267428 Solving 3.5 + n = -4.856267428 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = -4.856267428 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -4.856267428 + -3.5 n = -4.856267428 + -3.5 Combine like terms: -4.856267428 + -3.5 = -8.356267428 n = -8.356267428 Simplifying n = -8.356267428Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.356267428, -8.356267428}
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