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Simplifying 3n2 = -9n + 62 Reorder the terms: 3n2 = 62 + -9n Solving 3n2 = 62 + -9n Solving for variable 'n'. Reorder the terms: -62 + 9n + 3n2 = 62 + -9n + -62 + 9n Reorder the terms: -62 + 9n + 3n2 = 62 + -62 + -9n + 9n Combine like terms: 62 + -62 = 0 -62 + 9n + 3n2 = 0 + -9n + 9n -62 + 9n + 3n2 = -9n + 9n Combine like terms: -9n + 9n = 0 -62 + 9n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -20.66666667 + 3n + n2 = 0 Move the constant term to the right: Add '20.66666667' to each side of the equation. -20.66666667 + 3n + 20.66666667 + n2 = 0 + 20.66666667 Reorder the terms: -20.66666667 + 20.66666667 + 3n + n2 = 0 + 20.66666667 Combine like terms: -20.66666667 + 20.66666667 = 0.00000000 0.00000000 + 3n + n2 = 0 + 20.66666667 3n + n2 = 0 + 20.66666667 Combine like terms: 0 + 20.66666667 = 20.66666667 3n + n2 = 20.66666667 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 = 20.66666667 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 20.66666667 + 2.25 Combine like terms: 20.66666667 + 2.25 = 22.91666667 2.25 + 3n + n2 = 22.91666667 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 22.91666667 Calculate the square root of the right side: 4.787135539 Break this problem into two subproblems by setting (n + 1.5) equal to 4.787135539 and -4.787135539.Subproblem 1
n + 1.5 = 4.787135539 Simplifying n + 1.5 = 4.787135539 Reorder the terms: 1.5 + n = 4.787135539 Solving 1.5 + n = 4.787135539 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 = 4.787135539 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 4.787135539 + -1.5 n = 4.787135539 + -1.5 Combine like terms: 4.787135539 + -1.5 = 3.287135539 n = 3.287135539 Simplifying n = 3.287135539Subproblem 2
n + 1.5 = -4.787135539 Simplifying n + 1.5 = -4.787135539 Reorder the terms: 1.5 + n = -4.787135539 Solving 1.5 + n = -4.787135539 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 = -4.787135539 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -4.787135539 + -1.5 n = -4.787135539 + -1.5 Combine like terms: -4.787135539 + -1.5 = -6.287135539 n = -6.287135539 Simplifying n = -6.287135539Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.287135539, -6.287135539}
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