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Simplifying n2 + 13n + -22 = 7 Reorder the terms: -22 + 13n + n2 = 7 Solving -22 + 13n + n2 = 7 Solving for variable 'n'. Reorder the terms: -22 + -7 + 13n + n2 = 7 + -7 Combine like terms: -22 + -7 = -29 -29 + 13n + n2 = 7 + -7 Combine like terms: 7 + -7 = 0 -29 + 13n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '29' to each side of the equation. -29 + 13n + 29 + n2 = 0 + 29 Reorder the terms: -29 + 29 + 13n + n2 = 0 + 29 Combine like terms: -29 + 29 = 0 0 + 13n + n2 = 0 + 29 13n + n2 = 0 + 29 Combine like terms: 0 + 29 = 29 13n + n2 = 29 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 = 29 + 42.25 Reorder the terms: 42.25 + 13n + n2 = 29 + 42.25 Combine like terms: 29 + 42.25 = 71.25 42.25 + 13n + n2 = 71.25 Factor a perfect square on the left side: (n + 6.5)(n + 6.5) = 71.25 Calculate the square root of the right side: 8.440971508 Break this problem into two subproblems by setting (n + 6.5) equal to 8.440971508 and -8.440971508.Subproblem 1
n + 6.5 = 8.440971508 Simplifying n + 6.5 = 8.440971508 Reorder the terms: 6.5 + n = 8.440971508 Solving 6.5 + n = 8.440971508 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 = 8.440971508 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = 8.440971508 + -6.5 n = 8.440971508 + -6.5 Combine like terms: 8.440971508 + -6.5 = 1.940971508 n = 1.940971508 Simplifying n = 1.940971508Subproblem 2
n + 6.5 = -8.440971508 Simplifying n + 6.5 = -8.440971508 Reorder the terms: 6.5 + n = -8.440971508 Solving 6.5 + n = -8.440971508 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 = -8.440971508 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = -8.440971508 + -6.5 n = -8.440971508 + -6.5 Combine like terms: -8.440971508 + -6.5 = -14.940971508 n = -14.940971508 Simplifying n = -14.940971508Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.940971508, -14.940971508}
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