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Simplifying (2n + 1)(n + 3) + (2n + 1)(n + -2) = 0 Reorder the terms: (1 + 2n)(n + 3) + (2n + 1)(n + -2) = 0 Reorder the terms: (1 + 2n)(3 + n) + (2n + 1)(n + -2) = 0 Multiply (1 + 2n) * (3 + n) (1(3 + n) + 2n * (3 + n)) + (2n + 1)(n + -2) = 0 ((3 * 1 + n * 1) + 2n * (3 + n)) + (2n + 1)(n + -2) = 0 ((3 + 1n) + 2n * (3 + n)) + (2n + 1)(n + -2) = 0 (3 + 1n + (3 * 2n + n * 2n)) + (2n + 1)(n + -2) = 0 (3 + 1n + (6n + 2n2)) + (2n + 1)(n + -2) = 0 Combine like terms: 1n + 6n = 7n (3 + 7n + 2n2) + (2n + 1)(n + -2) = 0 Reorder the terms: 3 + 7n + 2n2 + (1 + 2n)(n + -2) = 0 Reorder the terms: 3 + 7n + 2n2 + (1 + 2n)(-2 + n) = 0 Multiply (1 + 2n) * (-2 + n) 3 + 7n + 2n2 + (1(-2 + n) + 2n * (-2 + n)) = 0 3 + 7n + 2n2 + ((-2 * 1 + n * 1) + 2n * (-2 + n)) = 0 3 + 7n + 2n2 + ((-2 + 1n) + 2n * (-2 + n)) = 0 3 + 7n + 2n2 + (-2 + 1n + (-2 * 2n + n * 2n)) = 0 3 + 7n + 2n2 + (-2 + 1n + (-4n + 2n2)) = 0 Combine like terms: 1n + -4n = -3n 3 + 7n + 2n2 + (-2 + -3n + 2n2) = 0 Reorder the terms: 3 + -2 + 7n + -3n + 2n2 + 2n2 = 0 Combine like terms: 3 + -2 = 1 1 + 7n + -3n + 2n2 + 2n2 = 0 Combine like terms: 7n + -3n = 4n 1 + 4n + 2n2 + 2n2 = 0 Combine like terms: 2n2 + 2n2 = 4n2 1 + 4n + 4n2 = 0 Solving 1 + 4n + 4n2 = 0 Solving for variable 'n'. Factor a trinomial. (1 + 2n)(1 + 2n) = 0Subproblem 1
Set the factor '(1 + 2n)' equal to zero and attempt to solve: Simplifying 1 + 2n = 0 Solving 1 + 2n = 0 Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + 2n = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 2n = 0 + -1 2n = 0 + -1 Combine like terms: 0 + -1 = -1 2n = -1 Divide each side by '2'. n = -0.5 Simplifying n = -0.5Subproblem 2
Set the factor '(1 + 2n)' equal to zero and attempt to solve: Simplifying 1 + 2n = 0 Solving 1 + 2n = 0 Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + 2n = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 2n = 0 + -1 2n = 0 + -1 Combine like terms: 0 + -1 = -1 2n = -1 Divide each side by '2'. n = -0.5 Simplifying n = -0.5Solution
n = {-0.5, -0.5}
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