x^2+x+16=420

Solution for x^2+x+16=420 equation:

Simplifying
x2 + x + 16 = 420

Reorder the terms:
16 + x + x2 = 420

Solving
16 + x + x2 = 420

Solving for variable 'x'.

Reorder the terms:
16 + -420 + x + x2 = 420 + -420

Combine like terms: 16 + -420 = -404
-404 + x + x2 = 420 + -420

Combine like terms: 420 + -420 = 0
-404 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '404' to each side of the equation.
-404 + x + 404 + x2 = 0 + 404

Reorder the terms:
-404 + 404 + x + x2 = 0 + 404

Combine like terms: -404 + 404 = 0
0 + x + x2 = 0 + 404
x + x2 = 0 + 404

Combine like terms: 0 + 404 = 404
x + x2 = 404

The x term is x.  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.
x + 0.25 + x2 = 404 + 0.25

Reorder the terms:
0.25 + x + x2 = 404 + 0.25

Combine like terms: 404 + 0.25 = 404.25
0.25 + x + x2 = 404.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 404.25

Calculate the square root of the right side: 20.105969263

Break this problem into two subproblems by setting 
(x + 0.5) equal to 20.105969263 and -20.105969263.

Subproblem 1
x + 0.5 = 20.105969263

Simplifying
x + 0.5 = 20.105969263

Reorder the terms:
0.5 + x = 20.105969263

Solving
0.5 + x = 20.105969263

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 20.105969263 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 20.105969263 + -0.5
x = 20.105969263 + -0.5

Combine like terms: 20.105969263 + -0.5 = 19.605969263
x = 19.605969263

Simplifying
x = 19.605969263

Subproblem 2
x + 0.5 = -20.105969263

Simplifying
x + 0.5 = -20.105969263

Reorder the terms:
0.5 + x = -20.105969263

Solving
0.5 + x = -20.105969263

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -20.105969263 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -20.105969263 + -0.5
x = -20.105969263 + -0.5

Combine like terms: -20.105969263 + -0.5 = -20.605969263
x = -20.605969263

Simplifying
x = -20.605969263

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {19.605969263, -20.605969263}