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Simplifying 9n2 + -11 = -9n Reorder the terms: -11 + 9n2 = -9n Solving -11 + 9n2 = -9n Solving for variable 'n'. Reorder the terms: -11 + 9n + 9n2 = -9n + 9n Combine like terms: -9n + 9n = 0 -11 + 9n + 9n2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.222222222 + n + n2 = 0 Move the constant term to the right: Add '1.222222222' to each side of the equation. -1.222222222 + n + 1.222222222 + n2 = 0 + 1.222222222 Reorder the terms: -1.222222222 + 1.222222222 + n + n2 = 0 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + n + n2 = 0 + 1.222222222 n + n2 = 0 + 1.222222222 Combine like terms: 0 + 1.222222222 = 1.222222222 n + n2 = 1.222222222 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + n2 = 1.222222222 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = 1.222222222 + 0.25 Combine like terms: 1.222222222 + 0.25 = 1.472222222 1.25 + n2 = 1.472222222 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 1.472222222 Calculate the square root of the right side: 1.213351648 Break this problem into two subproblems by setting (n + 0.5) equal to 1.213351648 and -1.213351648.Subproblem 1
n + 0.5 = 1.213351648 Simplifying n + 0.5 = 1.213351648 Reorder the terms: 0.5 + n = 1.213351648 Solving 0.5 + n = 1.213351648 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 1.213351648 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.213351648 + -0.5 n = 1.213351648 + -0.5 Combine like terms: 1.213351648 + -0.5 = 0.713351648 n = 0.713351648 Simplifying n = 0.713351648Subproblem 2
n + 0.5 = -1.213351648 Simplifying n + 0.5 = -1.213351648 Reorder the terms: 0.5 + n = -1.213351648 Solving 0.5 + n = -1.213351648 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -1.213351648 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.213351648 + -0.5 n = -1.213351648 + -0.5 Combine like terms: -1.213351648 + -0.5 = -1.713351648 n = -1.713351648 Simplifying n = -1.713351648Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.713351648, -1.713351648}
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