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Simplifying 2n(n + -9) = n + -3(n + 7) Reorder the terms: 2n(-9 + n) = n + -3(n + 7) (-9 * 2n + n * 2n) = n + -3(n + 7) (-18n + 2n2) = n + -3(n + 7) Reorder the terms: -18n + 2n2 = n + -3(7 + n) -18n + 2n2 = n + (7 * -3 + n * -3) -18n + 2n2 = n + (-21 + -3n) Reorder the terms: -18n + 2n2 = -21 + n + -3n Combine like terms: n + -3n = -2n -18n + 2n2 = -21 + -2n Solving -18n + 2n2 = -21 + -2n Solving for variable 'n'. Reorder the terms: 21 + -18n + 2n + 2n2 = -21 + -2n + 21 + 2n Combine like terms: -18n + 2n = -16n 21 + -16n + 2n2 = -21 + -2n + 21 + 2n Reorder the terms: 21 + -16n + 2n2 = -21 + 21 + -2n + 2n Combine like terms: -21 + 21 = 0 21 + -16n + 2n2 = 0 + -2n + 2n 21 + -16n + 2n2 = -2n + 2n Combine like terms: -2n + 2n = 0 21 + -16n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 10.5 + -8n + n2 = 0 Move the constant term to the right: Add '-10.5' to each side of the equation. 10.5 + -8n + -10.5 + n2 = 0 + -10.5 Reorder the terms: 10.5 + -10.5 + -8n + n2 = 0 + -10.5 Combine like terms: 10.5 + -10.5 = 0.0 0.0 + -8n + n2 = 0 + -10.5 -8n + n2 = 0 + -10.5 Combine like terms: 0 + -10.5 = -10.5 -8n + n2 = -10.5 The n term is -8n. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8n + 16 + n2 = -10.5 + 16 Reorder the terms: 16 + -8n + n2 = -10.5 + 16 Combine like terms: -10.5 + 16 = 5.5 16 + -8n + n2 = 5.5 Factor a perfect square on the left side: (n + -4)(n + -4) = 5.5 Calculate the square root of the right side: 2.34520788 Break this problem into two subproblems by setting (n + -4) equal to 2.34520788 and -2.34520788.Subproblem 1
n + -4 = 2.34520788 Simplifying n + -4 = 2.34520788 Reorder the terms: -4 + n = 2.34520788 Solving -4 + n = 2.34520788 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = 2.34520788 + 4 Combine like terms: -4 + 4 = 0 0 + n = 2.34520788 + 4 n = 2.34520788 + 4 Combine like terms: 2.34520788 + 4 = 6.34520788 n = 6.34520788 Simplifying n = 6.34520788Subproblem 2
n + -4 = -2.34520788 Simplifying n + -4 = -2.34520788 Reorder the terms: -4 + n = -2.34520788 Solving -4 + n = -2.34520788 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = -2.34520788 + 4 Combine like terms: -4 + 4 = 0 0 + n = -2.34520788 + 4 n = -2.34520788 + 4 Combine like terms: -2.34520788 + 4 = 1.65479212 n = 1.65479212 Simplifying n = 1.65479212Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.34520788, 1.65479212}
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