n^2+2n=33

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


Simplifying
n2 + 2n = 33

Reorder the terms:
2n + n2 = 33

Solving
2n + n2 = 33

Solving for variable 'n'.

Reorder the terms:
-33 + 2n + n2 = 33 + -33

Combine like terms: 33 + -33 = 0
-33 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '33' to each side of the equation.
-33 + 2n + 33 + n2 = 0 + 33

Reorder the terms:
-33 + 33 + 2n + n2 = 0 + 33

Combine like terms: -33 + 33 = 0
0 + 2n + n2 = 0 + 33
2n + n2 = 0 + 33

Combine like terms: 0 + 33 = 33
2n + n2 = 33

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 = 33 + 1

Reorder the terms:
1 + 2n + n2 = 33 + 1

Combine like terms: 33 + 1 = 34
1 + 2n + n2 = 34

Factor a perfect square on the left side:
(n + 1)(n + 1) = 34

Calculate the square root of the right side: 5.830951895

Break this problem into two subproblems by setting 
(n + 1) equal to 5.830951895 and -5.830951895.


Subproblem 1
n + 1 = 5.830951895

Simplifying
n + 1 = 5.830951895

Reorder the terms:
1 + n = 5.830951895

Solving
1 + n = 5.830951895

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 = 5.830951895 + -1

Combine like terms: 1 + -1 = 0
0 + n = 5.830951895 + -1
n = 5.830951895 + -1

Combine like terms: 5.830951895 + -1 = 4.830951895
n = 4.830951895

Simplifying
n = 4.830951895


Subproblem 2
n + 1 = -5.830951895

Simplifying
n + 1 = -5.830951895

Reorder the terms:
1 + n = -5.830951895

Solving
1 + n = -5.830951895

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 = -5.830951895 + -1

Combine like terms: 1 + -1 = 0
0 + n = -5.830951895 + -1
n = -5.830951895 + -1

Combine like terms: -5.830951895 + -1 = -6.830951895
n = -6.830951895

Simplifying
n = -6.830951895


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.830951895, -6.830951895}

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