x(x+1)=3x^2-2

Solution for x(x+1)=3x^2-2 equation:

Simplifying
x(x + 1) = 3x2 + -2

Reorder the terms:
x(1 + x) = 3x2 + -2
(1 * x + x * x) = 3x2 + -2
(1x + x2) = 3x2 + -2

Reorder the terms:
1x + x2 = -2 + 3x2

Solving
1x + x2 = -2 + 3x2

Solving for variable 'x'.

Reorder the terms:
2 + 1x + x2 + -3x2 = -2 + 3x2 + 2 + -3x2

Combine like terms: x2 + -3x2 = -2x2
2 + 1x + -2x2 = -2 + 3x2 + 2 + -3x2

Reorder the terms:
2 + 1x + -2x2 = -2 + 2 + 3x2 + -3x2

Combine like terms: -2 + 2 = 0
2 + 1x + -2x2 = 0 + 3x2 + -3x2
2 + 1x + -2x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
2 + 1x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-1 + -0.5x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.5x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.5x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.5x + x2 = 0 + 1
-0.5x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.5x + x2 = 1

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 1 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 1 + 0.0625

Combine like terms: 1 + 0.0625 = 1.0625
0.0625 + -0.5x + x2 = 1.0625

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(x + -0.25) equal to 1.030776406 and -1.030776406.

Subproblem 1
x + -0.25 = 1.030776406

Simplifying
x + -0.25 = 1.030776406

Reorder the terms:
-0.25 + x = 1.030776406

Solving
-0.25 + x = 1.030776406

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = 1.030776406 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = 1.030776406 + 0.25
x = 1.030776406 + 0.25

Combine like terms: 1.030776406 + 0.25 = 1.280776406
x = 1.280776406

Simplifying
x = 1.280776406

Subproblem 2
x + -0.25 = -1.030776406

Simplifying
x + -0.25 = -1.030776406

Reorder the terms:
-0.25 + x = -1.030776406

Solving
-0.25 + x = -1.030776406

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = -1.030776406 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = -1.030776406 + 0.25
x = -1.030776406 + 0.25

Combine like terms: -1.030776406 + 0.25 = -0.780776406
x = -0.780776406

Simplifying
x = -0.780776406

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.280776406, -0.780776406}