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Simplifying 3n2 + -54 = -3n2 + 1 + 12n Reorder the terms: -54 + 3n2 = -3n2 + 1 + 12n Reorder the terms: -54 + 3n2 = 1 + 12n + -3n2 Solving -54 + 3n2 = 1 + 12n + -3n2 Solving for variable 'n'. Reorder the terms: -54 + -1 + -12n + 3n2 + 3n2 = 1 + 12n + -3n2 + -1 + -12n + 3n2 Combine like terms: -54 + -1 = -55 -55 + -12n + 3n2 + 3n2 = 1 + 12n + -3n2 + -1 + -12n + 3n2 Combine like terms: 3n2 + 3n2 = 6n2 -55 + -12n + 6n2 = 1 + 12n + -3n2 + -1 + -12n + 3n2 Reorder the terms: -55 + -12n + 6n2 = 1 + -1 + 12n + -12n + -3n2 + 3n2 Combine like terms: 1 + -1 = 0 -55 + -12n + 6n2 = 0 + 12n + -12n + -3n2 + 3n2 -55 + -12n + 6n2 = 12n + -12n + -3n2 + 3n2 Combine like terms: 12n + -12n = 0 -55 + -12n + 6n2 = 0 + -3n2 + 3n2 -55 + -12n + 6n2 = -3n2 + 3n2 Combine like terms: -3n2 + 3n2 = 0 -55 + -12n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -9.166666667 + -2n + n2 = 0 Move the constant term to the right: Add '9.166666667' to each side of the equation. -9.166666667 + -2n + 9.166666667 + n2 = 0 + 9.166666667 Reorder the terms: -9.166666667 + 9.166666667 + -2n + n2 = 0 + 9.166666667 Combine like terms: -9.166666667 + 9.166666667 = 0.000000000 0.000000000 + -2n + n2 = 0 + 9.166666667 -2n + n2 = 0 + 9.166666667 Combine like terms: 0 + 9.166666667 = 9.166666667 -2n + n2 = 9.166666667 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 = 9.166666667 + 1 Reorder the terms: 1 + -2n + n2 = 9.166666667 + 1 Combine like terms: 9.166666667 + 1 = 10.166666667 1 + -2n + n2 = 10.166666667 Factor a perfect square on the left side: (n + -1)(n + -1) = 10.166666667 Calculate the square root of the right side: 3.188521078 Break this problem into two subproblems by setting (n + -1) equal to 3.188521078 and -3.188521078.Subproblem 1
n + -1 = 3.188521078 Simplifying n + -1 = 3.188521078 Reorder the terms: -1 + n = 3.188521078 Solving -1 + n = 3.188521078 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 = 3.188521078 + 1 Combine like terms: -1 + 1 = 0 0 + n = 3.188521078 + 1 n = 3.188521078 + 1 Combine like terms: 3.188521078 + 1 = 4.188521078 n = 4.188521078 Simplifying n = 4.188521078Subproblem 2
n + -1 = -3.188521078 Simplifying n + -1 = -3.188521078 Reorder the terms: -1 + n = -3.188521078 Solving -1 + n = -3.188521078 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 = -3.188521078 + 1 Combine like terms: -1 + 1 = 0 0 + n = -3.188521078 + 1 n = -3.188521078 + 1 Combine like terms: -3.188521078 + 1 = -2.188521078 n = -2.188521078 Simplifying n = -2.188521078Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.188521078, -2.188521078}
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