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Simplifying 2n(5n + 3) + 7(5n + -3) = 0 Reorder the terms: 2n(3 + 5n) + 7(5n + -3) = 0 (3 * 2n + 5n * 2n) + 7(5n + -3) = 0 (6n + 10n2) + 7(5n + -3) = 0 Reorder the terms: 6n + 10n2 + 7(-3 + 5n) = 0 6n + 10n2 + (-3 * 7 + 5n * 7) = 0 6n + 10n2 + (-21 + 35n) = 0 Reorder the terms: -21 + 6n + 35n + 10n2 = 0 Combine like terms: 6n + 35n = 41n -21 + 41n + 10n2 = 0 Solving -21 + 41n + 10n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -2.1 + 4.1n + n2 = 0.0 Move the constant term to the right: Add '2.1' to each side of the equation. -2.1 + 4.1n + 2.1 + n2 = 0.0 + 2.1 Reorder the terms: -2.1 + 2.1 + 4.1n + n2 = 0.0 + 2.1 Combine like terms: -2.1 + 2.1 = 0.0 0.0 + 4.1n + n2 = 0.0 + 2.1 4.1n + n2 = 0.0 + 2.1 Combine like terms: 0.0 + 2.1 = 2.1 4.1n + n2 = 2.1 The n term is 4.1n. Take half its coefficient (2.05). Square it (4.2025) and add it to both sides. Add '4.2025' to each side of the equation. 4.1n + 4.2025 + n2 = 2.1 + 4.2025 Reorder the terms: 4.2025 + 4.1n + n2 = 2.1 + 4.2025 Combine like terms: 2.1 + 4.2025 = 6.3025 4.2025 + 4.1n + n2 = 6.3025 Factor a perfect square on the left side: (n + 2.05)(n + 2.05) = 6.3025 Calculate the square root of the right side: 2.510478042 Break this problem into two subproblems by setting (n + 2.05) equal to 2.510478042 and -2.510478042.Subproblem 1
n + 2.05 = 2.510478042 Simplifying n + 2.05 = 2.510478042 Reorder the terms: 2.05 + n = 2.510478042 Solving 2.05 + n = 2.510478042 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + n = 2.510478042 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + n = 2.510478042 + -2.05 n = 2.510478042 + -2.05 Combine like terms: 2.510478042 + -2.05 = 0.460478042 n = 0.460478042 Simplifying n = 0.460478042Subproblem 2
n + 2.05 = -2.510478042 Simplifying n + 2.05 = -2.510478042 Reorder the terms: 2.05 + n = -2.510478042 Solving 2.05 + n = -2.510478042 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + n = -2.510478042 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + n = -2.510478042 + -2.05 n = -2.510478042 + -2.05 Combine like terms: -2.510478042 + -2.05 = -4.560478042 n = -4.560478042 Simplifying n = -4.560478042Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.460478042, -4.560478042}
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