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Simplifying n2 + 2n + -68 = 0 Reorder the terms: -68 + 2n + n2 = 0 Solving -68 + 2n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '68' to each side of the equation. -68 + 2n + 68 + n2 = 0 + 68 Reorder the terms: -68 + 68 + 2n + n2 = 0 + 68 Combine like terms: -68 + 68 = 0 0 + 2n + n2 = 0 + 68 2n + n2 = 0 + 68 Combine like terms: 0 + 68 = 68 2n + n2 = 68 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 68 + 1 Reorder the terms: 1 + 2n + n2 = 68 + 1 Combine like terms: 68 + 1 = 69 1 + 2n + n2 = 69 Factor a perfect square on the left side: (n + 1)(n + 1) = 69 Calculate the square root of the right side: 8.306623863 Break this problem into two subproblems by setting (n + 1) equal to 8.306623863 and -8.306623863.Subproblem 1
n + 1 = 8.306623863 Simplifying n + 1 = 8.306623863 Reorder the terms: 1 + n = 8.306623863 Solving 1 + n = 8.306623863 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 8.306623863 + -1 Combine like terms: 1 + -1 = 0 0 + n = 8.306623863 + -1 n = 8.306623863 + -1 Combine like terms: 8.306623863 + -1 = 7.306623863 n = 7.306623863 Simplifying n = 7.306623863Subproblem 2
n + 1 = -8.306623863 Simplifying n + 1 = -8.306623863 Reorder the terms: 1 + n = -8.306623863 Solving 1 + n = -8.306623863 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -8.306623863 + -1 Combine like terms: 1 + -1 = 0 0 + n = -8.306623863 + -1 n = -8.306623863 + -1 Combine like terms: -8.306623863 + -1 = -9.306623863 n = -9.306623863 Simplifying n = -9.306623863Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.306623863, -9.306623863}
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