5n^2+15n-46=0

Solution for 5n^2+15n-46=0 equation:

Simplifying
5n2 + 15n + -46 = 0

Reorder the terms:
-46 + 15n + 5n2 = 0

Solving
-46 + 15n + 5n2 = 0

Solving for variable 'n'.

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

Divide each side by '5'.
-9.2 + 3n + n2 = 0

Move the constant term to the right:

Add '9.2' to each side of the equation.
-9.2 + 3n + 9.2 + n2 = 0 + 9.2

Reorder the terms:
-9.2 + 9.2 + 3n + n2 = 0 + 9.2

Combine like terms: -9.2 + 9.2 = 0.0
0.0 + 3n + n2 = 0 + 9.2
3n + n2 = 0 + 9.2

Combine like terms: 0 + 9.2 = 9.2
3n + n2 = 9.2

The n term is 3n.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3n + 2.25 + n2 = 9.2 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 9.2 + 2.25

Combine like terms: 9.2 + 2.25 = 11.45
2.25 + 3n + n2 = 11.45

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 11.45

Calculate the square root of the right side: 3.383784863

Break this problem into two subproblems by setting 
(n + 1.5) equal to 3.383784863 and -3.383784863.

Subproblem 1
n + 1.5 = 3.383784863

Simplifying
n + 1.5 = 3.383784863

Reorder the terms:
1.5 + n = 3.383784863

Solving
1.5 + n = 3.383784863

Solving for variable 'n'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + n = 3.383784863 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + n = 3.383784863 + -1.5
n = 3.383784863 + -1.5

Combine like terms: 3.383784863 + -1.5 = 1.883784863
n = 1.883784863

Simplifying
n = 1.883784863

Subproblem 2
n + 1.5 = -3.383784863

Simplifying
n + 1.5 = -3.383784863

Reorder the terms:
1.5 + n = -3.383784863

Solving
1.5 + n = -3.383784863

Solving for variable 'n'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + n = -3.383784863 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + n = -3.383784863 + -1.5
n = -3.383784863 + -1.5

Combine like terms: -3.383784863 + -1.5 = -4.883784863
n = -4.883784863

Simplifying
n = -4.883784863

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.883784863, -4.883784863}