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Simplifying n2 + 13n + 27 = -5 Reorder the terms: 27 + 13n + n2 = -5 Solving 27 + 13n + n2 = -5 Solving for variable 'n'. Reorder the terms: 27 + 5 + 13n + n2 = -5 + 5 Combine like terms: 27 + 5 = 32 32 + 13n + n2 = -5 + 5 Combine like terms: -5 + 5 = 0 32 + 13n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-32' to each side of the equation. 32 + 13n + -32 + n2 = 0 + -32 Reorder the terms: 32 + -32 + 13n + n2 = 0 + -32 Combine like terms: 32 + -32 = 0 0 + 13n + n2 = 0 + -32 13n + n2 = 0 + -32 Combine like terms: 0 + -32 = -32 13n + n2 = -32 The n term is 13n. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13n + 42.25 + n2 = -32 + 42.25 Reorder the terms: 42.25 + 13n + n2 = -32 + 42.25 Combine like terms: -32 + 42.25 = 10.25 42.25 + 13n + n2 = 10.25 Factor a perfect square on the left side: (n + 6.5)(n + 6.5) = 10.25 Calculate the square root of the right side: 3.201562119 Break this problem into two subproblems by setting (n + 6.5) equal to 3.201562119 and -3.201562119.Subproblem 1
n + 6.5 = 3.201562119 Simplifying n + 6.5 = 3.201562119 Reorder the terms: 6.5 + n = 3.201562119 Solving 6.5 + n = 3.201562119 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + n = 3.201562119 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = 3.201562119 + -6.5 n = 3.201562119 + -6.5 Combine like terms: 3.201562119 + -6.5 = -3.298437881 n = -3.298437881 Simplifying n = -3.298437881Subproblem 2
n + 6.5 = -3.201562119 Simplifying n + 6.5 = -3.201562119 Reorder the terms: 6.5 + n = -3.201562119 Solving 6.5 + n = -3.201562119 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + n = -3.201562119 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = -3.201562119 + -6.5 n = -3.201562119 + -6.5 Combine like terms: -3.201562119 + -6.5 = -9.701562119 n = -9.701562119 Simplifying n = -9.701562119Solution
The solution to the problem is based on the solutions from the subproblems. n = {-3.298437881, -9.701562119}
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