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Simplifying 3n(5n + 9) = 11 Reorder the terms: 3n(9 + 5n) = 11 (9 * 3n + 5n * 3n) = 11 (27n + 15n2) = 11 Solving 27n + 15n2 = 11 Solving for variable 'n'. Reorder the terms: -11 + 27n + 15n2 = 11 + -11 Combine like terms: 11 + -11 = 0 -11 + 27n + 15n2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.7333333333 + 1.8n + n2 = 0 Move the constant term to the right: Add '0.7333333333' to each side of the equation. -0.7333333333 + 1.8n + 0.7333333333 + n2 = 0 + 0.7333333333 Reorder the terms: -0.7333333333 + 0.7333333333 + 1.8n + n2 = 0 + 0.7333333333 Combine like terms: -0.7333333333 + 0.7333333333 = 0.0000000000 0.0000000000 + 1.8n + n2 = 0 + 0.7333333333 1.8n + n2 = 0 + 0.7333333333 Combine like terms: 0 + 0.7333333333 = 0.7333333333 1.8n + n2 = 0.7333333333 The n term is 1.8n. Take half its coefficient (0.9). Square it (0.81) and add it to both sides. Add '0.81' to each side of the equation. 1.8n + 0.81 + n2 = 0.7333333333 + 0.81 Reorder the terms: 0.81 + 1.8n + n2 = 0.7333333333 + 0.81 Combine like terms: 0.7333333333 + 0.81 = 1.5433333333 0.81 + 1.8n + n2 = 1.5433333333 Factor a perfect square on the left side: (n + 0.9)(n + 0.9) = 1.5433333333 Calculate the square root of the right side: 1.242309677 Break this problem into two subproblems by setting (n + 0.9) equal to 1.242309677 and -1.242309677.Subproblem 1
n + 0.9 = 1.242309677 Simplifying n + 0.9 = 1.242309677 Reorder the terms: 0.9 + n = 1.242309677 Solving 0.9 + n = 1.242309677 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + n = 1.242309677 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + n = 1.242309677 + -0.9 n = 1.242309677 + -0.9 Combine like terms: 1.242309677 + -0.9 = 0.342309677 n = 0.342309677 Simplifying n = 0.342309677Subproblem 2
n + 0.9 = -1.242309677 Simplifying n + 0.9 = -1.242309677 Reorder the terms: 0.9 + n = -1.242309677 Solving 0.9 + n = -1.242309677 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + n = -1.242309677 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + n = -1.242309677 + -0.9 n = -1.242309677 + -0.9 Combine like terms: -1.242309677 + -0.9 = -2.142309677 n = -2.142309677 Simplifying n = -2.142309677Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.342309677, -2.142309677}
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