n(n)+n=71

Solution for n(n)+n=71 equation:

Simplifying
n(n) + n = 71

Multiply n * n
n2 + n = 71

Reorder the terms:
n + n2 = 71

Solving
n + n2 = 71

Solving for variable 'n'.

Reorder the terms:
-71 + n + n2 = 71 + -71

Combine like terms: 71 + -71 = 0
-71 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '71' to each side of the equation.
-71 + n + 71 + n2 = 0 + 71

Reorder the terms:
-71 + 71 + n + n2 = 0 + 71

Combine like terms: -71 + 71 = 0
0 + n + n2 = 0 + 71
n + n2 = 0 + 71

Combine like terms: 0 + 71 = 71
n + n2 = 71

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 = 71 + 0.25

Reorder the terms:
0.25 + n + n2 = 71 + 0.25

Combine like terms: 71 + 0.25 = 71.25
0.25 + n + n2 = 71.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 71.25

Calculate the square root of the right side: 8.440971508

Break this problem into two subproblems by setting 
(n + 0.5) equal to 8.440971508 and -8.440971508.

Subproblem 1
n + 0.5 = 8.440971508

Simplifying
n + 0.5 = 8.440971508

Reorder the terms:
0.5 + n = 8.440971508

Solving
0.5 + n = 8.440971508

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 = 8.440971508 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 8.440971508 + -0.5
n = 8.440971508 + -0.5

Combine like terms: 8.440971508 + -0.5 = 7.940971508
n = 7.940971508

Simplifying
n = 7.940971508

Subproblem 2
n + 0.5 = -8.440971508

Simplifying
n + 0.5 = -8.440971508

Reorder the terms:
0.5 + n = -8.440971508

Solving
0.5 + n = -8.440971508

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 = -8.440971508 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -8.440971508 + -0.5
n = -8.440971508 + -0.5

Combine like terms: -8.440971508 + -0.5 = -8.940971508
n = -8.940971508

Simplifying
n = -8.940971508

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {7.940971508, -8.940971508}