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Simplifying 7 + 6n(5 + 2n) = 7(2 + 8n) 7 + (5 * 6n + 2n * 6n) = 7(2 + 8n) 7 + (30n + 12n2) = 7(2 + 8n) 7 + 30n + 12n2 = (2 * 7 + 8n * 7) 7 + 30n + 12n2 = (14 + 56n) Solving 7 + 30n + 12n2 = 14 + 56n Solving for variable 'n'. Reorder the terms: 7 + -14 + 30n + -56n + 12n2 = 14 + 56n + -14 + -56n Combine like terms: 7 + -14 = -7 -7 + 30n + -56n + 12n2 = 14 + 56n + -14 + -56n Combine like terms: 30n + -56n = -26n -7 + -26n + 12n2 = 14 + 56n + -14 + -56n Reorder the terms: -7 + -26n + 12n2 = 14 + -14 + 56n + -56n Combine like terms: 14 + -14 = 0 -7 + -26n + 12n2 = 0 + 56n + -56n -7 + -26n + 12n2 = 56n + -56n Combine like terms: 56n + -56n = 0 -7 + -26n + 12n2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.5833333333 + -2.166666667n + n2 = 0 Move the constant term to the right: Add '0.5833333333' to each side of the equation. -0.5833333333 + -2.166666667n + 0.5833333333 + n2 = 0 + 0.5833333333 Reorder the terms: -0.5833333333 + 0.5833333333 + -2.166666667n + n2 = 0 + 0.5833333333 Combine like terms: -0.5833333333 + 0.5833333333 = 0.0000000000 0.0000000000 + -2.166666667n + n2 = 0 + 0.5833333333 -2.166666667n + n2 = 0 + 0.5833333333 Combine like terms: 0 + 0.5833333333 = 0.5833333333 -2.166666667n + n2 = 0.5833333333 The n term is -2.166666667n. Take half its coefficient (-1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. -2.166666667n + 1.173611113 + n2 = 0.5833333333 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667n + n2 = 0.5833333333 + 1.173611113 Combine like terms: 0.5833333333 + 1.173611113 = 1.7569444463 1.173611113 + -2.166666667n + n2 = 1.7569444463 Factor a perfect square on the left side: (n + -1.083333334)(n + -1.083333334) = 1.7569444463 Calculate the square root of the right side: 1.325497811 Break this problem into two subproblems by setting (n + -1.083333334) equal to 1.325497811 and -1.325497811.Subproblem 1
n + -1.083333334 = 1.325497811 Simplifying n + -1.083333334 = 1.325497811 Reorder the terms: -1.083333334 + n = 1.325497811 Solving -1.083333334 + n = 1.325497811 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + n = 1.325497811 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + n = 1.325497811 + 1.083333334 n = 1.325497811 + 1.083333334 Combine like terms: 1.325497811 + 1.083333334 = 2.408831145 n = 2.408831145 Simplifying n = 2.408831145Subproblem 2
n + -1.083333334 = -1.325497811 Simplifying n + -1.083333334 = -1.325497811 Reorder the terms: -1.083333334 + n = -1.325497811 Solving -1.083333334 + n = -1.325497811 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + n = -1.325497811 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + n = -1.325497811 + 1.083333334 n = -1.325497811 + 1.083333334 Combine like terms: -1.325497811 + 1.083333334 = -0.242164477 n = -0.242164477 Simplifying n = -0.242164477Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.408831145, -0.242164477}
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