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Simplifying n2 + -11 = 3 + -1n Reorder the terms: -11 + n2 = 3 + -1n Solving -11 + n2 = 3 + -1n Solving for variable 'n'. Reorder the terms: -11 + -3 + n + n2 = 3 + -1n + -3 + n Combine like terms: -11 + -3 = -14 -14 + n + n2 = 3 + -1n + -3 + n Reorder the terms: -14 + n + n2 = 3 + -3 + -1n + n Combine like terms: 3 + -3 = 0 -14 + n + n2 = 0 + -1n + n -14 + n + n2 = -1n + n Combine like terms: -1n + n = 0 -14 + n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '14' to each side of the equation. -14 + n + 14 + n2 = 0 + 14 Reorder the terms: -14 + 14 + n + n2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + n + n2 = 0 + 14 n + n2 = 0 + 14 Combine like terms: 0 + 14 = 14 n + n2 = 14 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. n + 0.25 + n2 = 14 + 0.25 Reorder the terms: 0.25 + n + n2 = 14 + 0.25 Combine like terms: 14 + 0.25 = 14.25 0.25 + n + n2 = 14.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 14.25 Calculate the square root of the right side: 3.774917218 Break this problem into two subproblems by setting (n + 0.5) equal to 3.774917218 and -3.774917218.Subproblem 1
n + 0.5 = 3.774917218 Simplifying n + 0.5 = 3.774917218 Reorder the terms: 0.5 + n = 3.774917218 Solving 0.5 + n = 3.774917218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 3.774917218 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 3.774917218 + -0.5 n = 3.774917218 + -0.5 Combine like terms: 3.774917218 + -0.5 = 3.274917218 n = 3.274917218 Simplifying n = 3.274917218Subproblem 2
n + 0.5 = -3.774917218 Simplifying n + 0.5 = -3.774917218 Reorder the terms: 0.5 + n = -3.774917218 Solving 0.5 + n = -3.774917218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -3.774917218 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -3.774917218 + -0.5 n = -3.774917218 + -0.5 Combine like terms: -3.774917218 + -0.5 = -4.274917218 n = -4.274917218 Simplifying n = -4.274917218Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.274917218, -4.274917218}
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