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Simplifying 3n(n + -1) = 5n + 3 + -3n Reorder the terms: 3n(-1 + n) = 5n + 3 + -3n (-1 * 3n + n * 3n) = 5n + 3 + -3n (-3n + 3n2) = 5n + 3 + -3n Reorder the terms: -3n + 3n2 = 3 + 5n + -3n Combine like terms: 5n + -3n = 2n -3n + 3n2 = 3 + 2n Solving -3n + 3n2 = 3 + 2n Solving for variable 'n'. Reorder the terms: -3 + -3n + -2n + 3n2 = 3 + 2n + -3 + -2n Combine like terms: -3n + -2n = -5n -3 + -5n + 3n2 = 3 + 2n + -3 + -2n Reorder the terms: -3 + -5n + 3n2 = 3 + -3 + 2n + -2n Combine like terms: 3 + -3 = 0 -3 + -5n + 3n2 = 0 + 2n + -2n -3 + -5n + 3n2 = 2n + -2n Combine like terms: 2n + -2n = 0 -3 + -5n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1 + -1.666666667n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -1.666666667n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + -1.666666667n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1.666666667n + n2 = 0 + 1 -1.666666667n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1.666666667n + n2 = 1 The n term is -1.666666667n. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667n + 0.6944444447 + n2 = 1 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667n + n2 = 1 + 0.6944444447 Combine like terms: 1 + 0.6944444447 = 1.6944444447 0.6944444447 + -1.666666667n + n2 = 1.6944444447 Factor a perfect square on the left side: (n + -0.8333333335)(n + -0.8333333335) = 1.6944444447 Calculate the square root of the right side: 1.301708279 Break this problem into two subproblems by setting (n + -0.8333333335) equal to 1.301708279 and -1.301708279.Subproblem 1
n + -0.8333333335 = 1.301708279 Simplifying n + -0.8333333335 = 1.301708279 Reorder the terms: -0.8333333335 + n = 1.301708279 Solving -0.8333333335 + n = 1.301708279 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + n = 1.301708279 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + n = 1.301708279 + 0.8333333335 n = 1.301708279 + 0.8333333335 Combine like terms: 1.301708279 + 0.8333333335 = 2.1350416125 n = 2.1350416125 Simplifying n = 2.1350416125Subproblem 2
n + -0.8333333335 = -1.301708279 Simplifying n + -0.8333333335 = -1.301708279 Reorder the terms: -0.8333333335 + n = -1.301708279 Solving -0.8333333335 + n = -1.301708279 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + n = -1.301708279 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + n = -1.301708279 + 0.8333333335 n = -1.301708279 + 0.8333333335 Combine like terms: -1.301708279 + 0.8333333335 = -0.4683749455 n = -0.4683749455 Simplifying n = -0.4683749455Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.1350416125, -0.4683749455}
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