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Simplifying 3n + -16 = n2 + -6n + 3 Reorder the terms: -16 + 3n = n2 + -6n + 3 Reorder the terms: -16 + 3n = 3 + -6n + n2 Solving -16 + 3n = 3 + -6n + n2 Solving for variable 'n'. Reorder the terms: -16 + -3 + 3n + 6n + -1n2 = 3 + -6n + n2 + -3 + 6n + -1n2 Combine like terms: -16 + -3 = -19 -19 + 3n + 6n + -1n2 = 3 + -6n + n2 + -3 + 6n + -1n2 Combine like terms: 3n + 6n = 9n -19 + 9n + -1n2 = 3 + -6n + n2 + -3 + 6n + -1n2 Reorder the terms: -19 + 9n + -1n2 = 3 + -3 + -6n + 6n + n2 + -1n2 Combine like terms: 3 + -3 = 0 -19 + 9n + -1n2 = 0 + -6n + 6n + n2 + -1n2 -19 + 9n + -1n2 = -6n + 6n + n2 + -1n2 Combine like terms: -6n + 6n = 0 -19 + 9n + -1n2 = 0 + n2 + -1n2 -19 + 9n + -1n2 = n2 + -1n2 Combine like terms: n2 + -1n2 = 0 -19 + 9n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 19 + -9n + n2 = 0 Move the constant term to the right: Add '-19' to each side of the equation. 19 + -9n + -19 + n2 = 0 + -19 Reorder the terms: 19 + -19 + -9n + n2 = 0 + -19 Combine like terms: 19 + -19 = 0 0 + -9n + n2 = 0 + -19 -9n + n2 = 0 + -19 Combine like terms: 0 + -19 = -19 -9n + n2 = -19 The n term is -9n. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9n + 20.25 + n2 = -19 + 20.25 Reorder the terms: 20.25 + -9n + n2 = -19 + 20.25 Combine like terms: -19 + 20.25 = 1.25 20.25 + -9n + n2 = 1.25 Factor a perfect square on the left side: (n + -4.5)(n + -4.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (n + -4.5) equal to 1.118033989 and -1.118033989.Subproblem 1
n + -4.5 = 1.118033989 Simplifying n + -4.5 = 1.118033989 Reorder the terms: -4.5 + n = 1.118033989 Solving -4.5 + n = 1.118033989 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + n = 1.118033989 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = 1.118033989 + 4.5 n = 1.118033989 + 4.5 Combine like terms: 1.118033989 + 4.5 = 5.618033989 n = 5.618033989 Simplifying n = 5.618033989Subproblem 2
n + -4.5 = -1.118033989 Simplifying n + -4.5 = -1.118033989 Reorder the terms: -4.5 + n = -1.118033989 Solving -4.5 + n = -1.118033989 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + n = -1.118033989 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = -1.118033989 + 4.5 n = -1.118033989 + 4.5 Combine like terms: -1.118033989 + 4.5 = 3.381966011 n = 3.381966011 Simplifying n = 3.381966011Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.618033989, 3.381966011}
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