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Simplifying -7n(n + 2) = -6n + -1 Reorder the terms: -7n(2 + n) = -6n + -1 (2 * -7n + n * -7n) = -6n + -1 (-14n + -7n2) = -6n + -1 Reorder the terms: -14n + -7n2 = -1 + -6n Solving -14n + -7n2 = -1 + -6n Solving for variable 'n'. Reorder the terms: 1 + -14n + 6n + -7n2 = -1 + -6n + 1 + 6n Combine like terms: -14n + 6n = -8n 1 + -8n + -7n2 = -1 + -6n + 1 + 6n Reorder the terms: 1 + -8n + -7n2 = -1 + 1 + -6n + 6n Combine like terms: -1 + 1 = 0 1 + -8n + -7n2 = 0 + -6n + 6n 1 + -8n + -7n2 = -6n + 6n Combine like terms: -6n + 6n = 0 1 + -8n + -7n2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -0.1428571429 + 1.142857143n + n2 = 0 Move the constant term to the right: Add '0.1428571429' to each side of the equation. -0.1428571429 + 1.142857143n + 0.1428571429 + n2 = 0 + 0.1428571429 Reorder the terms: -0.1428571429 + 0.1428571429 + 1.142857143n + n2 = 0 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + 1.142857143n + n2 = 0 + 0.1428571429 1.142857143n + n2 = 0 + 0.1428571429 Combine like terms: 0 + 0.1428571429 = 0.1428571429 1.142857143n + n2 = 0.1428571429 The n term is 1.142857143n. Take half its coefficient (0.5714285715). Square it (0.3265306123) and add it to both sides. Add '0.3265306123' to each side of the equation. 1.142857143n + 0.3265306123 + n2 = 0.1428571429 + 0.3265306123 Reorder the terms: 0.3265306123 + 1.142857143n + n2 = 0.1428571429 + 0.3265306123 Combine like terms: 0.1428571429 + 0.3265306123 = 0.4693877552 0.3265306123 + 1.142857143n + n2 = 0.4693877552 Factor a perfect square on the left side: (n + 0.5714285715)(n + 0.5714285715) = 0.4693877552 Calculate the square root of the right side: 0.685118789 Break this problem into two subproblems by setting (n + 0.5714285715) equal to 0.685118789 and -0.685118789.Subproblem 1
n + 0.5714285715 = 0.685118789 Simplifying n + 0.5714285715 = 0.685118789 Reorder the terms: 0.5714285715 + n = 0.685118789 Solving 0.5714285715 + n = 0.685118789 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + n = 0.685118789 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + n = 0.685118789 + -0.5714285715 n = 0.685118789 + -0.5714285715 Combine like terms: 0.685118789 + -0.5714285715 = 0.1136902175 n = 0.1136902175 Simplifying n = 0.1136902175Subproblem 2
n + 0.5714285715 = -0.685118789 Simplifying n + 0.5714285715 = -0.685118789 Reorder the terms: 0.5714285715 + n = -0.685118789 Solving 0.5714285715 + n = -0.685118789 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + n = -0.685118789 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + n = -0.685118789 + -0.5714285715 n = -0.685118789 + -0.5714285715 Combine like terms: -0.685118789 + -0.5714285715 = -1.2565473605 n = -1.2565473605 Simplifying n = -1.2565473605Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.1136902175, -1.2565473605}
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