5n^2=9n-1

Solution for 5n^2=9n-1 equation:

Simplifying
5n2 = 9n + -1

Reorder the terms:
5n2 = -1 + 9n

Solving
5n2 = -1 + 9n

Solving for variable 'n'.

Reorder the terms:
1 + -9n + 5n2 = -1 + 9n + 1 + -9n

Reorder the terms:
1 + -9n + 5n2 = -1 + 1 + 9n + -9n

Combine like terms: -1 + 1 = 0
1 + -9n + 5n2 = 0 + 9n + -9n
1 + -9n + 5n2 = 9n + -9n

Combine like terms: 9n + -9n = 0
1 + -9n + 5n2 = 0

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

Divide each side by '5'.
0.2 + -1.8n + n2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + -1.8n + -0.2 + n2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + -1.8n + n2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + -1.8n + n2 = 0 + -0.2
-1.8n + n2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
-1.8n + n2 = -0.2

The n term is -1.8n.  Take half its coefficient (-0.9).
Square it (0.81) and add it to both sides.

Add '0.81' to each side of the equation.
-1.8n + 0.81 + n2 = -0.2 + 0.81

Reorder the terms:
0.81 + -1.8n + n2 = -0.2 + 0.81

Combine like terms: -0.2 + 0.81 = 0.61
0.81 + -1.8n + n2 = 0.61

Factor a perfect square on the left side:
(n + -0.9)(n + -0.9) = 0.61

Calculate the square root of the right side: 0.781024968

Break this problem into two subproblems by setting 
(n + -0.9) equal to 0.781024968 and -0.781024968.

Subproblem 1
n + -0.9 = 0.781024968

Simplifying
n + -0.9 = 0.781024968

Reorder the terms:
-0.9 + n = 0.781024968

Solving
-0.9 + n = 0.781024968

Solving for variable 'n'.

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

Add '0.9' to each side of the equation.
-0.9 + 0.9 + n = 0.781024968 + 0.9

Combine like terms: -0.9 + 0.9 = 0.0
0.0 + n = 0.781024968 + 0.9
n = 0.781024968 + 0.9

Combine like terms: 0.781024968 + 0.9 = 1.681024968
n = 1.681024968

Simplifying
n = 1.681024968

Subproblem 2
n + -0.9 = -0.781024968

Simplifying
n + -0.9 = -0.781024968

Reorder the terms:
-0.9 + n = -0.781024968

Solving
-0.9 + n = -0.781024968

Solving for variable 'n'.

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

Add '0.9' to each side of the equation.
-0.9 + 0.9 + n = -0.781024968 + 0.9

Combine like terms: -0.9 + 0.9 = 0.0
0.0 + n = -0.781024968 + 0.9
n = -0.781024968 + 0.9

Combine like terms: -0.781024968 + 0.9 = 0.118975032
n = 0.118975032

Simplifying
n = 0.118975032

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.681024968, 0.118975032}