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Simplifying n2 + -9n + 11 = 0 Reorder the terms: 11 + -9n + n2 = 0 Solving 11 + -9n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '-11' to each side of the equation. 11 + -9n + -11 + n2 = 0 + -11 Reorder the terms: 11 + -11 + -9n + n2 = 0 + -11 Combine like terms: 11 + -11 = 0 0 + -9n + n2 = 0 + -11 -9n + n2 = 0 + -11 Combine like terms: 0 + -11 = -11 -9n + n2 = -11 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 = -11 + 20.25 Reorder the terms: 20.25 + -9n + n2 = -11 + 20.25 Combine like terms: -11 + 20.25 = 9.25 20.25 + -9n + n2 = 9.25 Factor a perfect square on the left side: (n + -4.5)(n + -4.5) = 9.25 Calculate the square root of the right side: 3.041381265 Break this problem into two subproblems by setting (n + -4.5) equal to 3.041381265 and -3.041381265.Subproblem 1
n + -4.5 = 3.041381265 Simplifying n + -4.5 = 3.041381265 Reorder the terms: -4.5 + n = 3.041381265 Solving -4.5 + n = 3.041381265 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 = 3.041381265 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = 3.041381265 + 4.5 n = 3.041381265 + 4.5 Combine like terms: 3.041381265 + 4.5 = 7.541381265 n = 7.541381265 Simplifying n = 7.541381265Subproblem 2
n + -4.5 = -3.041381265 Simplifying n + -4.5 = -3.041381265 Reorder the terms: -4.5 + n = -3.041381265 Solving -4.5 + n = -3.041381265 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 = -3.041381265 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = -3.041381265 + 4.5 n = -3.041381265 + 4.5 Combine like terms: -3.041381265 + 4.5 = 1.458618735 n = 1.458618735 Simplifying n = 1.458618735Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.541381265, 1.458618735}
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