2x(x+1)=6

Solution for 2x(x+1)=6 equation:

Simplifying
2x(x + 1) = 6

Reorder the terms:
2x(1 + x) = 6
(1 * 2x + x * 2x) = 6
(2x + 2x2) = 6

Solving
2x + 2x2 = 6

Solving for variable 'x'.

Reorder the terms:
-6 + 2x + 2x2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + 2x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-3 + x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-3 + x + x2)' equal to zero and attempt to solve:

Simplifying
-3 + x + x2 = 0

Solving
-3 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + x + x2 = 0 + 3

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

Combine like terms: 0 + 3 = 3
x + x2 = 3

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 = 3 + 0.25

Reorder the terms:
0.25 + x + x2 = 3 + 0.25

Combine like terms: 3 + 0.25 = 3.25
0.25 + x + x2 = 3.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
x + 0.5 = 1.802775638

Simplifying
x + 0.5 = 1.802775638

Reorder the terms:
0.5 + x = 1.802775638

Solving
0.5 + x = 1.802775638

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 = 1.802775638 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 1.802775638 + -0.5
x = 1.802775638 + -0.5

Combine like terms: 1.802775638 + -0.5 = 1.302775638
x = 1.302775638

Simplifying
x = 1.302775638

Subproblem 2
x + 0.5 = -1.802775638

Simplifying
x + 0.5 = -1.802775638

Reorder the terms:
0.5 + x = -1.802775638

Solving
0.5 + x = -1.802775638

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 = -1.802775638 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -1.802775638 + -0.5
x = -1.802775638 + -0.5

Combine like terms: -1.802775638 + -0.5 = -2.302775638
x = -2.302775638

Simplifying
x = -2.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.302775638, -2.302775638}
Solution
x = {1.302775638, -2.302775638}