10n^2-20n-45=0

Solution for 10n^2-20n-45=0 equation:

Simplifying
10n2 + -20n + -45 = 0

Reorder the terms:
-45 + -20n + 10n2 = 0

Solving
-45 + -20n + 10n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '5'.
5(-9 + -4n + 2n2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor '(-9 + -4n + 2n2)' equal to zero and attempt to solve:

Simplifying
-9 + -4n + 2n2 = 0

Solving
-9 + -4n + 2n2 = 0

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

Divide each side by '2'.
-4.5 + -2n + n2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + -2n + 4.5 + n2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + -2n + n2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + -2n + n2 = 0 + 4.5
-2n + n2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
-2n + n2 = 4.5

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 4.5 + 1

Reorder the terms:
1 + -2n + n2 = 4.5 + 1

Combine like terms: 4.5 + 1 = 5.5
1 + -2n + n2 = 5.5

Factor a perfect square on the left side:
(n + -1)(n + -1) = 5.5

Calculate the square root of the right side: 2.34520788

Break this problem into two subproblems by setting 
(n + -1) equal to 2.34520788 and -2.34520788.

Subproblem 1
n + -1 = 2.34520788

Simplifying
n + -1 = 2.34520788

Reorder the terms:
-1 + n = 2.34520788

Solving
-1 + n = 2.34520788

Solving for variable 'n'.

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

Add '1' to each side of the equation.
-1 + 1 + n = 2.34520788 + 1

Combine like terms: -1 + 1 = 0
0 + n = 2.34520788 + 1
n = 2.34520788 + 1

Combine like terms: 2.34520788 + 1 = 3.34520788
n = 3.34520788

Simplifying
n = 3.34520788

Subproblem 2
n + -1 = -2.34520788

Simplifying
n + -1 = -2.34520788

Reorder the terms:
-1 + n = -2.34520788

Solving
-1 + n = -2.34520788

Solving for variable 'n'.

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

Add '1' to each side of the equation.
-1 + 1 + n = -2.34520788 + 1

Combine like terms: -1 + 1 = 0
0 + n = -2.34520788 + 1
n = -2.34520788 + 1

Combine like terms: -2.34520788 + 1 = -1.34520788
n = -1.34520788

Simplifying
n = -1.34520788

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {3.34520788, -1.34520788}
Solution
n = {3.34520788, -1.34520788}