7n^2+6n-23=0

Solution for 7n^2+6n-23=0 equation:

Simplifying
7n2 + 6n + -23 = 0

Reorder the terms:
-23 + 6n + 7n2 = 0

Solving
-23 + 6n + 7n2 = 0

Solving for variable 'n'.

Begin completing the square.  Divide all terms by
7 the coefficient of the squared term: 

Divide each side by '7'.
-3.285714286 + 0.8571428571n + n2 = 0

Move the constant term to the right:

Add '3.285714286' to each side of the equation.
-3.285714286 + 0.8571428571n + 3.285714286 + n2 = 0 + 3.285714286

Reorder the terms:
-3.285714286 + 3.285714286 + 0.8571428571n + n2 = 0 + 3.285714286

Combine like terms: -3.285714286 + 3.285714286 = 0.000000000
0.000000000 + 0.8571428571n + n2 = 0 + 3.285714286
0.8571428571n + n2 = 0 + 3.285714286

Combine like terms: 0 + 3.285714286 = 3.285714286
0.8571428571n + n2 = 3.285714286

The n term is 0.8571428571n.  Take half its coefficient (0.4285714286).
Square it (0.1836734694) and add it to both sides.

Add '0.1836734694' to each side of the equation.
0.8571428571n + 0.1836734694 + n2 = 3.285714286 + 0.1836734694

Reorder the terms:
0.1836734694 + 0.8571428571n + n2 = 3.285714286 + 0.1836734694

Combine like terms: 3.285714286 + 0.1836734694 = 3.4693877554
0.1836734694 + 0.8571428571n + n2 = 3.4693877554

Factor a perfect square on the left side:
(n + 0.4285714286)(n + 0.4285714286) = 3.4693877554

Calculate the square root of the right side: 1.862629259

Break this problem into two subproblems by setting 
(n + 0.4285714286) equal to 1.862629259 and -1.862629259.

Subproblem 1
n + 0.4285714286 = 1.862629259

Simplifying
n + 0.4285714286 = 1.862629259

Reorder the terms:
0.4285714286 + n = 1.862629259

Solving
0.4285714286 + n = 1.862629259

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-0.4285714286' to each side of the equation.
0.4285714286 + -0.4285714286 + n = 1.862629259 + -0.4285714286

Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000
0.0000000000 + n = 1.862629259 + -0.4285714286
n = 1.862629259 + -0.4285714286

Combine like terms: 1.862629259 + -0.4285714286 = 1.4340578304
n = 1.4340578304

Simplifying
n = 1.4340578304

Subproblem 2
n + 0.4285714286 = -1.862629259

Simplifying
n + 0.4285714286 = -1.862629259

Reorder the terms:
0.4285714286 + n = -1.862629259

Solving
0.4285714286 + n = -1.862629259

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-0.4285714286' to each side of the equation.
0.4285714286 + -0.4285714286 + n = -1.862629259 + -0.4285714286

Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000
0.0000000000 + n = -1.862629259 + -0.4285714286
n = -1.862629259 + -0.4285714286

Combine like terms: -1.862629259 + -0.4285714286 = -2.2912006876
n = -2.2912006876

Simplifying
n = -2.2912006876

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.4340578304, -2.2912006876}