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Simplifying -7n(n + 6) = -6n + -8 Reorder the terms: -7n(6 + n) = -6n + -8 (6 * -7n + n * -7n) = -6n + -8 (-42n + -7n2) = -6n + -8 Reorder the terms: -42n + -7n2 = -8 + -6n Solving -42n + -7n2 = -8 + -6n Solving for variable 'n'. Reorder the terms: 8 + -42n + 6n + -7n2 = -8 + -6n + 8 + 6n Combine like terms: -42n + 6n = -36n 8 + -36n + -7n2 = -8 + -6n + 8 + 6n Reorder the terms: 8 + -36n + -7n2 = -8 + 8 + -6n + 6n Combine like terms: -8 + 8 = 0 8 + -36n + -7n2 = 0 + -6n + 6n 8 + -36n + -7n2 = -6n + 6n Combine like terms: -6n + 6n = 0 8 + -36n + -7n2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -1.142857143 + 5.142857143n + n2 = 0 Move the constant term to the right: Add '1.142857143' to each side of the equation. -1.142857143 + 5.142857143n + 1.142857143 + n2 = 0 + 1.142857143 Reorder the terms: -1.142857143 + 1.142857143 + 5.142857143n + n2 = 0 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + 5.142857143n + n2 = 0 + 1.142857143 5.142857143n + n2 = 0 + 1.142857143 Combine like terms: 0 + 1.142857143 = 1.142857143 5.142857143n + n2 = 1.142857143 The n term is 5.142857143n. Take half its coefficient (2.571428572). Square it (6.612244901) and add it to both sides. Add '6.612244901' to each side of the equation. 5.142857143n + 6.612244901 + n2 = 1.142857143 + 6.612244901 Reorder the terms: 6.612244901 + 5.142857143n + n2 = 1.142857143 + 6.612244901 Combine like terms: 1.142857143 + 6.612244901 = 7.755102044 6.612244901 + 5.142857143n + n2 = 7.755102044 Factor a perfect square on the left side: (n + 2.571428572)(n + 2.571428572) = 7.755102044 Calculate the square root of the right side: 2.784798385 Break this problem into two subproblems by setting (n + 2.571428572) equal to 2.784798385 and -2.784798385.Subproblem 1
n + 2.571428572 = 2.784798385 Simplifying n + 2.571428572 = 2.784798385 Reorder the terms: 2.571428572 + n = 2.784798385 Solving 2.571428572 + n = 2.784798385 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.571428572' to each side of the equation. 2.571428572 + -2.571428572 + n = 2.784798385 + -2.571428572 Combine like terms: 2.571428572 + -2.571428572 = 0.000000000 0.000000000 + n = 2.784798385 + -2.571428572 n = 2.784798385 + -2.571428572 Combine like terms: 2.784798385 + -2.571428572 = 0.213369813 n = 0.213369813 Simplifying n = 0.213369813Subproblem 2
n + 2.571428572 = -2.784798385 Simplifying n + 2.571428572 = -2.784798385 Reorder the terms: 2.571428572 + n = -2.784798385 Solving 2.571428572 + n = -2.784798385 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.571428572' to each side of the equation. 2.571428572 + -2.571428572 + n = -2.784798385 + -2.571428572 Combine like terms: 2.571428572 + -2.571428572 = 0.000000000 0.000000000 + n = -2.784798385 + -2.571428572 n = -2.784798385 + -2.571428572 Combine like terms: -2.784798385 + -2.571428572 = -5.356226957 n = -5.356226957 Simplifying n = -5.356226957Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.213369813, -5.356226957}
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