n^2+3n=33

Solution for n^2+3n=33 equation:

Simplifying
n2 + 3n = 33

Reorder the terms:
3n + n2 = 33

Solving
3n + n2 = 33

Solving for variable 'n'.

Reorder the terms:
-33 + 3n + n2 = 33 + -33

Combine like terms: 33 + -33 = 0
-33 + 3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '33' to each side of the equation.
-33 + 3n + 33 + n2 = 0 + 33

Reorder the terms:
-33 + 33 + 3n + n2 = 0 + 33

Combine like terms: -33 + 33 = 0
0 + 3n + n2 = 0 + 33
3n + n2 = 0 + 33

Combine like terms: 0 + 33 = 33
3n + n2 = 33

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 = 33 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 33 + 2.25

Combine like terms: 33 + 2.25 = 35.25
2.25 + 3n + n2 = 35.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 35.25

Calculate the square root of the right side: 5.937171044

Break this problem into two subproblems by setting 
(n + 1.5) equal to 5.937171044 and -5.937171044.

Subproblem 1
n + 1.5 = 5.937171044

Simplifying
n + 1.5 = 5.937171044

Reorder the terms:
1.5 + n = 5.937171044

Solving
1.5 + n = 5.937171044

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 = 5.937171044 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + n = 5.937171044 + -1.5
n = 5.937171044 + -1.5

Combine like terms: 5.937171044 + -1.5 = 4.437171044
n = 4.437171044

Simplifying
n = 4.437171044

Subproblem 2
n + 1.5 = -5.937171044

Simplifying
n + 1.5 = -5.937171044

Reorder the terms:
1.5 + n = -5.937171044

Solving
1.5 + n = -5.937171044

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 = -5.937171044 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + n = -5.937171044 + -1.5
n = -5.937171044 + -1.5

Combine like terms: -5.937171044 + -1.5 = -7.437171044
n = -7.437171044

Simplifying
n = -7.437171044

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.437171044, -7.437171044}