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Simplifying n2 + -6n + -9 = 11 Reorder the terms: -9 + -6n + n2 = 11 Solving -9 + -6n + n2 = 11 Solving for variable 'n'. Reorder the terms: -9 + -11 + -6n + n2 = 11 + -11 Combine like terms: -9 + -11 = -20 -20 + -6n + n2 = 11 + -11 Combine like terms: 11 + -11 = 0 -20 + -6n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '20' to each side of the equation. -20 + -6n + 20 + n2 = 0 + 20 Reorder the terms: -20 + 20 + -6n + n2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + -6n + n2 = 0 + 20 -6n + n2 = 0 + 20 Combine like terms: 0 + 20 = 20 -6n + n2 = 20 The n term is -6n. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6n + 9 + n2 = 20 + 9 Reorder the terms: 9 + -6n + n2 = 20 + 9 Combine like terms: 20 + 9 = 29 9 + -6n + n2 = 29 Factor a perfect square on the left side: (n + -3)(n + -3) = 29 Calculate the square root of the right side: 5.385164807 Break this problem into two subproblems by setting (n + -3) equal to 5.385164807 and -5.385164807.Subproblem 1
n + -3 = 5.385164807 Simplifying n + -3 = 5.385164807 Reorder the terms: -3 + n = 5.385164807 Solving -3 + n = 5.385164807 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 5.385164807 + 3 Combine like terms: -3 + 3 = 0 0 + n = 5.385164807 + 3 n = 5.385164807 + 3 Combine like terms: 5.385164807 + 3 = 8.385164807 n = 8.385164807 Simplifying n = 8.385164807Subproblem 2
n + -3 = -5.385164807 Simplifying n + -3 = -5.385164807 Reorder the terms: -3 + n = -5.385164807 Solving -3 + n = -5.385164807 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -5.385164807 + 3 Combine like terms: -3 + 3 = 0 0 + n = -5.385164807 + 3 n = -5.385164807 + 3 Combine like terms: -5.385164807 + 3 = -2.385164807 n = -2.385164807 Simplifying n = -2.385164807Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.385164807, -2.385164807}
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