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Simplifying 7n2 + -8n = 14 Reorder the terms: -8n + 7n2 = 14 Solving -8n + 7n2 = 14 Solving for variable 'n'. Reorder the terms: -14 + -8n + 7n2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + -8n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2 + -1.142857143n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -1.142857143n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + -1.142857143n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -1.142857143n + n2 = 0 + 2 -1.142857143n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 -1.142857143n + n2 = 2 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 = 2 + 0.3265306123 Reorder the terms: 0.3265306123 + -1.142857143n + n2 = 2 + 0.3265306123 Combine like terms: 2 + 0.3265306123 = 2.3265306123 0.3265306123 + -1.142857143n + n2 = 2.3265306123 Factor a perfect square on the left side: (n + -0.5714285715)(n + -0.5714285715) = 2.3265306123 Calculate the square root of the right side: 1.525296893 Break this problem into two subproblems by setting (n + -0.5714285715) equal to 1.525296893 and -1.525296893.Subproblem 1
n + -0.5714285715 = 1.525296893 Simplifying n + -0.5714285715 = 1.525296893 Reorder the terms: -0.5714285715 + n = 1.525296893 Solving -0.5714285715 + n = 1.525296893 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 = 1.525296893 + 0.5714285715 Combine like terms: -0.5714285715 + 0.5714285715 = 0.0000000000 0.0000000000 + n = 1.525296893 + 0.5714285715 n = 1.525296893 + 0.5714285715 Combine like terms: 1.525296893 + 0.5714285715 = 2.0967254645 n = 2.0967254645 Simplifying n = 2.0967254645Subproblem 2
n + -0.5714285715 = -1.525296893 Simplifying n + -0.5714285715 = -1.525296893 Reorder the terms: -0.5714285715 + n = -1.525296893 Solving -0.5714285715 + n = -1.525296893 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 = -1.525296893 + 0.5714285715 Combine like terms: -0.5714285715 + 0.5714285715 = 0.0000000000 0.0000000000 + n = -1.525296893 + 0.5714285715 n = -1.525296893 + 0.5714285715 Combine like terms: -1.525296893 + 0.5714285715 = -0.9538683215 n = -0.9538683215 Simplifying n = -0.9538683215Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.0967254645, -0.9538683215}
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