n*(n+3)+22=10(n+2)

Solution for n*(n+3)+22=10(n+2) equation:

Simplifying
n(n + 3) + 22 = 10(n + 2)

Reorder the terms:
n(3 + n) + 22 = 10(n + 2)
(3 * n + n * n) + 22 = 10(n + 2)
(3n + n2) + 22 = 10(n + 2)

Reorder the terms:
22 + 3n + n2 = 10(n + 2)

Reorder the terms:
22 + 3n + n2 = 10(2 + n)
22 + 3n + n2 = (2 * 10 + n * 10)
22 + 3n + n2 = (20 + 10n)

Solving
22 + 3n + n2 = 20 + 10n

Solving for variable 'n'.

Reorder the terms:
22 + -20 + 3n + -10n + n2 = 20 + 10n + -20 + -10n

Combine like terms: 22 + -20 = 2
2 + 3n + -10n + n2 = 20 + 10n + -20 + -10n

Combine like terms: 3n + -10n = -7n
2 + -7n + n2 = 20 + 10n + -20 + -10n

Reorder the terms:
2 + -7n + n2 = 20 + -20 + 10n + -10n

Combine like terms: 20 + -20 = 0
2 + -7n + n2 = 0 + 10n + -10n
2 + -7n + n2 = 10n + -10n

Combine like terms: 10n + -10n = 0
2 + -7n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -7n + -2 + n2 = 0 + -2

Reorder the terms:
2 + -2 + -7n + n2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -7n + n2 = 0 + -2
-7n + n2 = 0 + -2

Combine like terms: 0 + -2 = -2
-7n + n2 = -2

The n term is -7n.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7n + 12.25 + n2 = -2 + 12.25

Reorder the terms:
12.25 + -7n + n2 = -2 + 12.25

Combine like terms: -2 + 12.25 = 10.25
12.25 + -7n + n2 = 10.25

Factor a perfect square on the left side:
(n + -3.5)(n + -3.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(n + -3.5) equal to 3.201562119 and -3.201562119.

Subproblem 1
n + -3.5 = 3.201562119

Simplifying
n + -3.5 = 3.201562119

Reorder the terms:
-3.5 + n = 3.201562119

Solving
-3.5 + n = 3.201562119

Solving for variable 'n'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + n = 3.201562119 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + n = 3.201562119 + 3.5
n = 3.201562119 + 3.5

Combine like terms: 3.201562119 + 3.5 = 6.701562119
n = 6.701562119

Simplifying
n = 6.701562119

Subproblem 2
n + -3.5 = -3.201562119

Simplifying
n + -3.5 = -3.201562119

Reorder the terms:
-3.5 + n = -3.201562119

Solving
-3.5 + n = -3.201562119

Solving for variable 'n'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + n = -3.201562119 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + n = -3.201562119 + 3.5
n = -3.201562119 + 3.5

Combine like terms: -3.201562119 + 3.5 = 0.298437881
n = 0.298437881

Simplifying
n = 0.298437881

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {6.701562119, 0.298437881}