6n^2-7n+5=9

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Solution for 6n^2-7n+5=9 equation: 


Simplifying
6n2 + -7n + 5 = 9

Reorder the terms:
5 + -7n + 6n2 = 9

Solving
5 + -7n + 6n2 = 9

Solving for variable 'n'.

Reorder the terms:
5 + -9 + -7n + 6n2 = 9 + -9

Combine like terms: 5 + -9 = -4
-4 + -7n + 6n2 = 9 + -9

Combine like terms: 9 + -9 = 0
-4 + -7n + 6n2 = 0

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

Divide each side by '6'.
-0.6666666667 + -1.166666667n + n2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -1.166666667n + 0.6666666667 + n2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -1.166666667n + n2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -1.166666667n + n2 = 0 + 0.6666666667
-1.166666667n + n2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-1.166666667n + n2 = 0.6666666667

The n term is -1.166666667n.  Take half its coefficient (-0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
-1.166666667n + 0.3402777780 + n2 = 0.6666666667 + 0.3402777780

Reorder the terms:
0.3402777780 + -1.166666667n + n2 = 0.6666666667 + 0.3402777780

Combine like terms: 0.6666666667 + 0.3402777780 = 1.0069444447
0.3402777780 + -1.166666667n + n2 = 1.0069444447

Factor a perfect square on the left side:
(n + -0.5833333335)(n + -0.5833333335) = 1.0069444447

Calculate the square root of the right side: 1.003466215

Break this problem into two subproblems by setting 
(n + -0.5833333335) equal to 1.003466215 and -1.003466215.


Subproblem 1
n + -0.5833333335 = 1.003466215

Simplifying
n + -0.5833333335 = 1.003466215

Reorder the terms:
-0.5833333335 + n = 1.003466215

Solving
-0.5833333335 + n = 1.003466215

Solving for variable 'n'.

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

Add '0.5833333335' to each side of the equation.
-0.5833333335 + 0.5833333335 + n = 1.003466215 + 0.5833333335

Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000
0.0000000000 + n = 1.003466215 + 0.5833333335
n = 1.003466215 + 0.5833333335

Combine like terms: 1.003466215 + 0.5833333335 = 1.5867995485
n = 1.5867995485

Simplifying
n = 1.5867995485


Subproblem 2
n + -0.5833333335 = -1.003466215

Simplifying
n + -0.5833333335 = -1.003466215

Reorder the terms:
-0.5833333335 + n = -1.003466215

Solving
-0.5833333335 + n = -1.003466215

Solving for variable 'n'.

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

Add '0.5833333335' to each side of the equation.
-0.5833333335 + 0.5833333335 + n = -1.003466215 + 0.5833333335

Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000
0.0000000000 + n = -1.003466215 + 0.5833333335
n = -1.003466215 + 0.5833333335

Combine like terms: -1.003466215 + 0.5833333335 = -0.4201328815
n = -0.4201328815

Simplifying
n = -0.4201328815


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.5867995485, -0.4201328815}

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