N^2+N-22=0

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Solution for N^2+N-22=0 equation: 


Simplifying
N2 + N + -22 = 0

Reorder the terms:
-22 + N + N2 = 0

Solving
-22 + N + N2 = 0

Solving for variable 'N'.

Begin completing the square.

Move the constant term to the right:

Add '22' to each side of the equation.
-22 + N + 22 + N2 = 0 + 22

Reorder the terms:
-22 + 22 + N + N2 = 0 + 22

Combine like terms: -22 + 22 = 0
0 + N + N2 = 0 + 22
N + N2 = 0 + 22

Combine like terms: 0 + 22 = 22
N + N2 = 22

The N term is N.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
N + 0.25 + N2 = 22 + 0.25

Reorder the terms:
0.25 + N + N2 = 22 + 0.25

Combine like terms: 22 + 0.25 = 22.25
0.25 + N + N2 = 22.25

Factor a perfect square on the left side:
(N + 0.5)(N + 0.5) = 22.25

Calculate the square root of the right side: 4.716990566

Break this problem into two subproblems by setting 
(N + 0.5) equal to 4.716990566 and -4.716990566.


Subproblem 1
N + 0.5 = 4.716990566

Simplifying
N + 0.5 = 4.716990566

Reorder the terms:
0.5 + N = 4.716990566

Solving
0.5 + N = 4.716990566

Solving for variable 'N'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + N = 4.716990566 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + N = 4.716990566 + -0.5
N = 4.716990566 + -0.5

Combine like terms: 4.716990566 + -0.5 = 4.216990566
N = 4.216990566

Simplifying
N = 4.216990566


Subproblem 2
N + 0.5 = -4.716990566

Simplifying
N + 0.5 = -4.716990566

Reorder the terms:
0.5 + N = -4.716990566

Solving
0.5 + N = -4.716990566

Solving for variable 'N'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + N = -4.716990566 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + N = -4.716990566 + -0.5
N = -4.716990566 + -0.5

Combine like terms: -4.716990566 + -0.5 = -5.216990566
N = -5.216990566

Simplifying
N = -5.216990566


Solution
The solution to the problem is based on the solutions
from the subproblems.
N = {4.216990566, -5.216990566}

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