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Simplifying 5n(2 + n) = 3n + 18 (2 * 5n + n * 5n) = 3n + 18 (10n + 5n2) = 3n + 18 Reorder the terms: 10n + 5n2 = 18 + 3n Solving 10n + 5n2 = 18 + 3n Solving for variable 'n'. Reorder the terms: -18 + 10n + -3n + 5n2 = 18 + 3n + -18 + -3n Combine like terms: 10n + -3n = 7n -18 + 7n + 5n2 = 18 + 3n + -18 + -3n Reorder the terms: -18 + 7n + 5n2 = 18 + -18 + 3n + -3n Combine like terms: 18 + -18 = 0 -18 + 7n + 5n2 = 0 + 3n + -3n -18 + 7n + 5n2 = 3n + -3n Combine like terms: 3n + -3n = 0 -18 + 7n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -3.6 + 1.4n + n2 = 0 Move the constant term to the right: Add '3.6' to each side of the equation. -3.6 + 1.4n + 3.6 + n2 = 0 + 3.6 Reorder the terms: -3.6 + 3.6 + 1.4n + n2 = 0 + 3.6 Combine like terms: -3.6 + 3.6 = 0.0 0.0 + 1.4n + n2 = 0 + 3.6 1.4n + n2 = 0 + 3.6 Combine like terms: 0 + 3.6 = 3.6 1.4n + n2 = 3.6 The n term is 1.4n. Take half its coefficient (0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. 1.4n + 0.49 + n2 = 3.6 + 0.49 Reorder the terms: 0.49 + 1.4n + n2 = 3.6 + 0.49 Combine like terms: 3.6 + 0.49 = 4.09 0.49 + 1.4n + n2 = 4.09 Factor a perfect square on the left side: (n + 0.7)(n + 0.7) = 4.09 Calculate the square root of the right side: 2.022374842 Break this problem into two subproblems by setting (n + 0.7) equal to 2.022374842 and -2.022374842.Subproblem 1
n + 0.7 = 2.022374842 Simplifying n + 0.7 = 2.022374842 Reorder the terms: 0.7 + n = 2.022374842 Solving 0.7 + n = 2.022374842 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + n = 2.022374842 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + n = 2.022374842 + -0.7 n = 2.022374842 + -0.7 Combine like terms: 2.022374842 + -0.7 = 1.322374842 n = 1.322374842 Simplifying n = 1.322374842Subproblem 2
n + 0.7 = -2.022374842 Simplifying n + 0.7 = -2.022374842 Reorder the terms: 0.7 + n = -2.022374842 Solving 0.7 + n = -2.022374842 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + n = -2.022374842 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + n = -2.022374842 + -0.7 n = -2.022374842 + -0.7 Combine like terms: -2.022374842 + -0.7 = -2.722374842 n = -2.722374842 Simplifying n = -2.722374842Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.322374842, -2.722374842}
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