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Simplifying (3n2 + 6n) + -1(-2n2 + -4n + 2) = 0 Reorder the terms: (6n + 3n2) + -1(-2n2 + -4n + 2) = 0 Remove parenthesis around (6n + 3n2) 6n + 3n2 + -1(-2n2 + -4n + 2) = 0 Reorder the terms: 6n + 3n2 + -1(2 + -4n + -2n2) = 0 6n + 3n2 + (2 * -1 + -4n * -1 + -2n2 * -1) = 0 6n + 3n2 + (-2 + 4n + 2n2) = 0 Reorder the terms: -2 + 6n + 4n + 3n2 + 2n2 = 0 Combine like terms: 6n + 4n = 10n -2 + 10n + 3n2 + 2n2 = 0 Combine like terms: 3n2 + 2n2 = 5n2 -2 + 10n + 5n2 = 0 Solving -2 + 10n + 5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.4 + 2n + n2 = 0 Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + 2n + 0.4 + n2 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + 2n + n2 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + 2n + n2 = 0 + 0.4 2n + n2 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 2n + n2 = 0.4 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 0.4 + 1 Reorder the terms: 1 + 2n + n2 = 0.4 + 1 Combine like terms: 0.4 + 1 = 1.4 1 + 2n + n2 = 1.4 Factor a perfect square on the left side: (n + 1)(n + 1) = 1.4 Calculate the square root of the right side: 1.183215957 Break this problem into two subproblems by setting (n + 1) equal to 1.183215957 and -1.183215957.Subproblem 1
n + 1 = 1.183215957 Simplifying n + 1 = 1.183215957 Reorder the terms: 1 + n = 1.183215957 Solving 1 + n = 1.183215957 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 1.183215957 + -1 Combine like terms: 1 + -1 = 0 0 + n = 1.183215957 + -1 n = 1.183215957 + -1 Combine like terms: 1.183215957 + -1 = 0.183215957 n = 0.183215957 Simplifying n = 0.183215957Subproblem 2
n + 1 = -1.183215957 Simplifying n + 1 = -1.183215957 Reorder the terms: 1 + n = -1.183215957 Solving 1 + n = -1.183215957 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -1.183215957 + -1 Combine like terms: 1 + -1 = 0 0 + n = -1.183215957 + -1 n = -1.183215957 + -1 Combine like terms: -1.183215957 + -1 = -2.183215957 n = -2.183215957 Simplifying n = -2.183215957Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.183215957, -2.183215957}
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