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Simplifying 10n2 + 9n = 11 Reorder the terms: 9n + 10n2 = 11 Solving 9n + 10n2 = 11 Solving for variable 'n'. Reorder the terms: -11 + 9n + 10n2 = 11 + -11 Combine like terms: 11 + -11 = 0 -11 + 9n + 10n2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -1.1 + 0.9n + n2 = 0 Move the constant term to the right: Add '1.1' to each side of the equation. -1.1 + 0.9n + 1.1 + n2 = 0 + 1.1 Reorder the terms: -1.1 + 1.1 + 0.9n + n2 = 0 + 1.1 Combine like terms: -1.1 + 1.1 = 0.0 0.0 + 0.9n + n2 = 0 + 1.1 0.9n + n2 = 0 + 1.1 Combine like terms: 0 + 1.1 = 1.1 0.9n + n2 = 1.1 The n term is 0.9n. Take half its coefficient (0.45). Square it (0.2025) and add it to both sides. Add '0.2025' to each side of the equation. 0.9n + 0.2025 + n2 = 1.1 + 0.2025 Reorder the terms: 0.2025 + 0.9n + n2 = 1.1 + 0.2025 Combine like terms: 1.1 + 0.2025 = 1.3025 0.2025 + 0.9n + n2 = 1.3025 Factor a perfect square on the left side: (n + 0.45)(n + 0.45) = 1.3025 Calculate the square root of the right side: 1.141271221 Break this problem into two subproblems by setting (n + 0.45) equal to 1.141271221 and -1.141271221.Subproblem 1
n + 0.45 = 1.141271221 Simplifying n + 0.45 = 1.141271221 Reorder the terms: 0.45 + n = 1.141271221 Solving 0.45 + n = 1.141271221 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + n = 1.141271221 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + n = 1.141271221 + -0.45 n = 1.141271221 + -0.45 Combine like terms: 1.141271221 + -0.45 = 0.691271221 n = 0.691271221 Simplifying n = 0.691271221Subproblem 2
n + 0.45 = -1.141271221 Simplifying n + 0.45 = -1.141271221 Reorder the terms: 0.45 + n = -1.141271221 Solving 0.45 + n = -1.141271221 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + n = -1.141271221 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + n = -1.141271221 + -0.45 n = -1.141271221 + -0.45 Combine like terms: -1.141271221 + -0.45 = -1.591271221 n = -1.591271221 Simplifying n = -1.591271221Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.691271221, -1.591271221}
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