10n^2+20n-46=0

Solution for 10n^2+20n-46=0 equation:

Simplifying
10n2 + 20n + -46 = 0

Reorder the terms:
-46 + 20n + 10n2 = 0

Solving
-46 + 20n + 10n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-23 + 10n + 5n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-23 + 10n + 5n2)' equal to zero and attempt to solve:

Simplifying
-23 + 10n + 5n2 = 0

Solving
-23 + 10n + 5n2 = 0

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

Divide each side by '5'.
-4.6 + 2n + n2 = 0

Move the constant term to the right:

Add '4.6' to each side of the equation.
-4.6 + 2n + 4.6 + n2 = 0 + 4.6

Reorder the terms:
-4.6 + 4.6 + 2n + n2 = 0 + 4.6

Combine like terms: -4.6 + 4.6 = 0.0
0.0 + 2n + n2 = 0 + 4.6
2n + n2 = 0 + 4.6

Combine like terms: 0 + 4.6 = 4.6
2n + n2 = 4.6

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 = 4.6 + 1

Reorder the terms:
1 + 2n + n2 = 4.6 + 1

Combine like terms: 4.6 + 1 = 5.6
1 + 2n + n2 = 5.6

Factor a perfect square on the left side:
(n + 1)(n + 1) = 5.6

Calculate the square root of the right side: 2.366431913

Break this problem into two subproblems by setting 
(n + 1) equal to 2.366431913 and -2.366431913.

Subproblem 1
n + 1 = 2.366431913

Simplifying
n + 1 = 2.366431913

Reorder the terms:
1 + n = 2.366431913

Solving
1 + n = 2.366431913

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 = 2.366431913 + -1

Combine like terms: 1 + -1 = 0
0 + n = 2.366431913 + -1
n = 2.366431913 + -1

Combine like terms: 2.366431913 + -1 = 1.366431913
n = 1.366431913

Simplifying
n = 1.366431913

Subproblem 2
n + 1 = -2.366431913

Simplifying
n + 1 = -2.366431913

Reorder the terms:
1 + n = -2.366431913

Solving
1 + n = -2.366431913

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 = -2.366431913 + -1

Combine like terms: 1 + -1 = 0
0 + n = -2.366431913 + -1
n = -2.366431913 + -1

Combine like terms: -2.366431913 + -1 = -3.366431913
n = -3.366431913

Simplifying
n = -3.366431913

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.366431913, -3.366431913}
Solution
n = {1.366431913, -3.366431913}