5n(n+2)=10

Solution for 5n(n+2)=10 equation:

Simplifying
5n(n + 2) = 10

Reorder the terms:
5n(2 + n) = 10
(2 * 5n + n * 5n) = 10
(10n + 5n2) = 10

Solving
10n + 5n2 = 10

Solving for variable 'n'.

Reorder the terms:
-10 + 10n + 5n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 10n + 5n2 = 0

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

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

Simplifying
-2 + 2n + n2 = 0

Solving
-2 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 2 + 1

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

Combine like terms: 2 + 1 = 3
1 + 2n + n2 = 3

Factor a perfect square on the left side:
(n + 1)(n + 1) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(n + 1) equal to 1.732050808 and -1.732050808.

Subproblem 1
n + 1 = 1.732050808

Simplifying
n + 1 = 1.732050808

Reorder the terms:
1 + n = 1.732050808

Solving
1 + n = 1.732050808

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 = 1.732050808 + -1

Combine like terms: 1 + -1 = 0
0 + n = 1.732050808 + -1
n = 1.732050808 + -1

Combine like terms: 1.732050808 + -1 = 0.732050808
n = 0.732050808

Simplifying
n = 0.732050808

Subproblem 2
n + 1 = -1.732050808

Simplifying
n + 1 = -1.732050808

Reorder the terms:
1 + n = -1.732050808

Solving
1 + n = -1.732050808

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 = -1.732050808 + -1

Combine like terms: 1 + -1 = 0
0 + n = -1.732050808 + -1
n = -1.732050808 + -1

Combine like terms: -1.732050808 + -1 = -2.732050808
n = -2.732050808

Simplifying
n = -2.732050808

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.732050808, -2.732050808}
Solution
n = {0.732050808, -2.732050808}