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Simplifying 13n2 = -10 + 25n + 2n2 Solving 13n2 = -10 + 25n + 2n2 Solving for variable 'n'. Reorder the terms: 10 + -25n + 13n2 + -2n2 = -10 + 25n + 2n2 + 10 + -25n + -2n2 Combine like terms: 13n2 + -2n2 = 11n2 10 + -25n + 11n2 = -10 + 25n + 2n2 + 10 + -25n + -2n2 Reorder the terms: 10 + -25n + 11n2 = -10 + 10 + 25n + -25n + 2n2 + -2n2 Combine like terms: -10 + 10 = 0 10 + -25n + 11n2 = 0 + 25n + -25n + 2n2 + -2n2 10 + -25n + 11n2 = 25n + -25n + 2n2 + -2n2 Combine like terms: 25n + -25n = 0 10 + -25n + 11n2 = 0 + 2n2 + -2n2 10 + -25n + 11n2 = 2n2 + -2n2 Combine like terms: 2n2 + -2n2 = 0 10 + -25n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. 0.9090909091 + -2.272727273n + n2 = 0 Move the constant term to the right: Add '-0.9090909091' to each side of the equation. 0.9090909091 + -2.272727273n + -0.9090909091 + n2 = 0 + -0.9090909091 Reorder the terms: 0.9090909091 + -0.9090909091 + -2.272727273n + n2 = 0 + -0.9090909091 Combine like terms: 0.9090909091 + -0.9090909091 = 0.0000000000 0.0000000000 + -2.272727273n + n2 = 0 + -0.9090909091 -2.272727273n + n2 = 0 + -0.9090909091 Combine like terms: 0 + -0.9090909091 = -0.9090909091 -2.272727273n + n2 = -0.9090909091 The n term is -2.272727273n. Take half its coefficient (-1.136363637). Square it (1.291322316) and add it to both sides. Add '1.291322316' to each side of the equation. -2.272727273n + 1.291322316 + n2 = -0.9090909091 + 1.291322316 Reorder the terms: 1.291322316 + -2.272727273n + n2 = -0.9090909091 + 1.291322316 Combine like terms: -0.9090909091 + 1.291322316 = 0.3822314069 1.291322316 + -2.272727273n + n2 = 0.3822314069 Factor a perfect square on the left side: (n + -1.136363637)(n + -1.136363637) = 0.3822314069 Calculate the square root of the right side: 0.618248661 Break this problem into two subproblems by setting (n + -1.136363637) equal to 0.618248661 and -0.618248661.Subproblem 1
n + -1.136363637 = 0.618248661 Simplifying n + -1.136363637 = 0.618248661 Reorder the terms: -1.136363637 + n = 0.618248661 Solving -1.136363637 + n = 0.618248661 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.136363637' to each side of the equation. -1.136363637 + 1.136363637 + n = 0.618248661 + 1.136363637 Combine like terms: -1.136363637 + 1.136363637 = 0.000000000 0.000000000 + n = 0.618248661 + 1.136363637 n = 0.618248661 + 1.136363637 Combine like terms: 0.618248661 + 1.136363637 = 1.754612298 n = 1.754612298 Simplifying n = 1.754612298Subproblem 2
n + -1.136363637 = -0.618248661 Simplifying n + -1.136363637 = -0.618248661 Reorder the terms: -1.136363637 + n = -0.618248661 Solving -1.136363637 + n = -0.618248661 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.136363637' to each side of the equation. -1.136363637 + 1.136363637 + n = -0.618248661 + 1.136363637 Combine like terms: -1.136363637 + 1.136363637 = 0.000000000 0.000000000 + n = -0.618248661 + 1.136363637 n = -0.618248661 + 1.136363637 Combine like terms: -0.618248661 + 1.136363637 = 0.518114976 n = 0.518114976 Simplifying n = 0.518114976Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.754612298, 0.518114976}
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