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Simplifying 5(1 + -1n) + 3n = 3n(n + -1) + -5n (1 * 5 + -1n * 5) + 3n = 3n(n + -1) + -5n (5 + -5n) + 3n = 3n(n + -1) + -5n Combine like terms: -5n + 3n = -2n 5 + -2n = 3n(n + -1) + -5n Reorder the terms: 5 + -2n = 3n(-1 + n) + -5n 5 + -2n = (-1 * 3n + n * 3n) + -5n 5 + -2n = (-3n + 3n2) + -5n Reorder the terms: 5 + -2n = -3n + -5n + 3n2 Combine like terms: -3n + -5n = -8n 5 + -2n = -8n + 3n2 Solving 5 + -2n = -8n + 3n2 Solving for variable 'n'. Combine like terms: -2n + 8n = 6n 5 + 6n + -3n2 = -8n + 3n2 + 8n + -3n2 Reorder the terms: 5 + 6n + -3n2 = -8n + 8n + 3n2 + -3n2 Combine like terms: -8n + 8n = 0 5 + 6n + -3n2 = 0 + 3n2 + -3n2 5 + 6n + -3n2 = 3n2 + -3n2 Combine like terms: 3n2 + -3n2 = 0 5 + 6n + -3n2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -1.666666667 + -2n + n2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + -2n + 1.666666667 + n2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + -2n + n2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + -2n + n2 = 0 + 1.666666667 -2n + n2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 -2n + n2 = 1.666666667 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 = 1.666666667 + 1 Reorder the terms: 1 + -2n + n2 = 1.666666667 + 1 Combine like terms: 1.666666667 + 1 = 2.666666667 1 + -2n + n2 = 2.666666667 Factor a perfect square on the left side: (n + -1)(n + -1) = 2.666666667 Calculate the square root of the right side: 1.632993162 Break this problem into two subproblems by setting (n + -1) equal to 1.632993162 and -1.632993162.Subproblem 1
n + -1 = 1.632993162 Simplifying n + -1 = 1.632993162 Reorder the terms: -1 + n = 1.632993162 Solving -1 + n = 1.632993162 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 = 1.632993162 + 1 Combine like terms: -1 + 1 = 0 0 + n = 1.632993162 + 1 n = 1.632993162 + 1 Combine like terms: 1.632993162 + 1 = 2.632993162 n = 2.632993162 Simplifying n = 2.632993162Subproblem 2
n + -1 = -1.632993162 Simplifying n + -1 = -1.632993162 Reorder the terms: -1 + n = -1.632993162 Solving -1 + n = -1.632993162 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 = -1.632993162 + 1 Combine like terms: -1 + 1 = 0 0 + n = -1.632993162 + 1 n = -1.632993162 + 1 Combine like terms: -1.632993162 + 1 = -0.632993162 n = -0.632993162 Simplifying n = -0.632993162Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.632993162, -0.632993162}
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