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Simplifying (n + 7)(n + 7) = -2n Reorder the terms: (7 + n)(n + 7) = -2n Reorder the terms: (7 + n)(7 + n) = -2n Multiply (7 + n) * (7 + n) (7(7 + n) + n(7 + n)) = -2n ((7 * 7 + n * 7) + n(7 + n)) = -2n ((49 + 7n) + n(7 + n)) = -2n (49 + 7n + (7 * n + n * n)) = -2n (49 + 7n + (7n + n2)) = -2n Combine like terms: 7n + 7n = 14n (49 + 14n + n2) = -2n Solving 49 + 14n + n2 = -2n Solving for variable 'n'. Reorder the terms: 49 + 14n + 2n + n2 = -2n + 2n Combine like terms: 14n + 2n = 16n 49 + 16n + n2 = -2n + 2n Combine like terms: -2n + 2n = 0 49 + 16n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-49' to each side of the equation. 49 + 16n + -49 + n2 = 0 + -49 Reorder the terms: 49 + -49 + 16n + n2 = 0 + -49 Combine like terms: 49 + -49 = 0 0 + 16n + n2 = 0 + -49 16n + n2 = 0 + -49 Combine like terms: 0 + -49 = -49 16n + n2 = -49 The n term is 16n. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16n + 64 + n2 = -49 + 64 Reorder the terms: 64 + 16n + n2 = -49 + 64 Combine like terms: -49 + 64 = 15 64 + 16n + n2 = 15 Factor a perfect square on the left side: (n + 8)(n + 8) = 15 Calculate the square root of the right side: 3.872983346 Break this problem into two subproblems by setting (n + 8) equal to 3.872983346 and -3.872983346.Subproblem 1
n + 8 = 3.872983346 Simplifying n + 8 = 3.872983346 Reorder the terms: 8 + n = 3.872983346 Solving 8 + n = 3.872983346 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + n = 3.872983346 + -8 Combine like terms: 8 + -8 = 0 0 + n = 3.872983346 + -8 n = 3.872983346 + -8 Combine like terms: 3.872983346 + -8 = -4.127016654 n = -4.127016654 Simplifying n = -4.127016654Subproblem 2
n + 8 = -3.872983346 Simplifying n + 8 = -3.872983346 Reorder the terms: 8 + n = -3.872983346 Solving 8 + n = -3.872983346 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + n = -3.872983346 + -8 Combine like terms: 8 + -8 = 0 0 + n = -3.872983346 + -8 n = -3.872983346 + -8 Combine like terms: -3.872983346 + -8 = -11.872983346 n = -11.872983346 Simplifying n = -11.872983346Solution
The solution to the problem is based on the solutions from the subproblems. n = {-4.127016654, -11.872983346}
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