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Simplifying 25n2 + 4n + -1 = 1 + -5n Reorder the terms: -1 + 4n + 25n2 = 1 + -5n Solving -1 + 4n + 25n2 = 1 + -5n Solving for variable 'n'. Reorder the terms: -1 + -1 + 4n + 5n + 25n2 = 1 + -5n + -1 + 5n Combine like terms: -1 + -1 = -2 -2 + 4n + 5n + 25n2 = 1 + -5n + -1 + 5n Combine like terms: 4n + 5n = 9n -2 + 9n + 25n2 = 1 + -5n + -1 + 5n Reorder the terms: -2 + 9n + 25n2 = 1 + -1 + -5n + 5n Combine like terms: 1 + -1 = 0 -2 + 9n + 25n2 = 0 + -5n + 5n -2 + 9n + 25n2 = -5n + 5n Combine like terms: -5n + 5n = 0 -2 + 9n + 25n2 = 0 Begin completing the square. Divide all terms by 25 the coefficient of the squared term: Divide each side by '25'. -0.08 + 0.36n + n2 = 0 Move the constant term to the right: Add '0.08' to each side of the equation. -0.08 + 0.36n + 0.08 + n2 = 0 + 0.08 Reorder the terms: -0.08 + 0.08 + 0.36n + n2 = 0 + 0.08 Combine like terms: -0.08 + 0.08 = 0.00 0.00 + 0.36n + n2 = 0 + 0.08 0.36n + n2 = 0 + 0.08 Combine like terms: 0 + 0.08 = 0.08 0.36n + n2 = 0.08 The n term is 0.36n. Take half its coefficient (0.18). Square it (0.0324) and add it to both sides. Add '0.0324' to each side of the equation. 0.36n + 0.0324 + n2 = 0.08 + 0.0324 Reorder the terms: 0.0324 + 0.36n + n2 = 0.08 + 0.0324 Combine like terms: 0.08 + 0.0324 = 0.1124 0.0324 + 0.36n + n2 = 0.1124 Factor a perfect square on the left side: (n + 0.18)(n + 0.18) = 0.1124 Calculate the square root of the right side: 0.335261092 Break this problem into two subproblems by setting (n + 0.18) equal to 0.335261092 and -0.335261092.Subproblem 1
n + 0.18 = 0.335261092 Simplifying n + 0.18 = 0.335261092 Reorder the terms: 0.18 + n = 0.335261092 Solving 0.18 + n = 0.335261092 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.18' to each side of the equation. 0.18 + -0.18 + n = 0.335261092 + -0.18 Combine like terms: 0.18 + -0.18 = 0.00 0.00 + n = 0.335261092 + -0.18 n = 0.335261092 + -0.18 Combine like terms: 0.335261092 + -0.18 = 0.155261092 n = 0.155261092 Simplifying n = 0.155261092Subproblem 2
n + 0.18 = -0.335261092 Simplifying n + 0.18 = -0.335261092 Reorder the terms: 0.18 + n = -0.335261092 Solving 0.18 + n = -0.335261092 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.18' to each side of the equation. 0.18 + -0.18 + n = -0.335261092 + -0.18 Combine like terms: 0.18 + -0.18 = 0.00 0.00 + n = -0.335261092 + -0.18 n = -0.335261092 + -0.18 Combine like terms: -0.335261092 + -0.18 = -0.515261092 n = -0.515261092 Simplifying n = -0.515261092Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.155261092, -0.515261092}
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