3x^2+3x+1=2048383

Solution for 3x^2+3x+1=2048383 equation:

Simplifying
3x2 + 3x + 1 = 2048383

Reorder the terms:
1 + 3x + 3x2 = 2048383

Solving
1 + 3x + 3x2 = 2048383

Solving for variable 'x'.

Reorder the terms:
1 + -2048383 + 3x + 3x2 = 2048383 + -2048383

Combine like terms: 1 + -2048383 = -2048382
-2048382 + 3x + 3x2 = 2048383 + -2048383

Combine like terms: 2048383 + -2048383 = 0
-2048382 + 3x + 3x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-682794 + x + x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-682794 + x + x2)' equal to zero and attempt to solve:

Simplifying
-682794 + x + x2 = 0

Solving
-682794 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '682794' to each side of the equation.
-682794 + x + 682794 + x2 = 0 + 682794

Reorder the terms:
-682794 + 682794 + x + x2 = 0 + 682794

Combine like terms: -682794 + 682794 = 0
0 + x + x2 = 0 + 682794
x + x2 = 0 + 682794

Combine like terms: 0 + 682794 = 682794
x + x2 = 682794

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 = 682794 + 0.25

Reorder the terms:
0.25 + x + x2 = 682794 + 0.25

Combine like terms: 682794 + 0.25 = 682794.25
0.25 + x + x2 = 682794.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 682794.25

Calculate the square root of the right side: 826.313651104

Break this problem into two subproblems by setting 
(x + 0.5) equal to 826.313651104 and -826.313651104.

Subproblem 1
x + 0.5 = 826.313651104

Simplifying
x + 0.5 = 826.313651104

Reorder the terms:
0.5 + x = 826.313651104

Solving
0.5 + x = 826.313651104

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 = 826.313651104 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 826.313651104 + -0.5
x = 826.313651104 + -0.5

Combine like terms: 826.313651104 + -0.5 = 825.813651104
x = 825.813651104

Simplifying
x = 825.813651104

Subproblem 2
x + 0.5 = -826.313651104

Simplifying
x + 0.5 = -826.313651104

Reorder the terms:
0.5 + x = -826.313651104

Solving
0.5 + x = -826.313651104

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 = -826.313651104 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -826.313651104 + -0.5
x = -826.313651104 + -0.5

Combine like terms: -826.313651104 + -0.5 = -826.813651104
x = -826.813651104

Simplifying
x = -826.813651104

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {825.813651104, -826.813651104}
Solution
x = {825.813651104, -826.813651104}