3x(x+1)=42

Solution for 3x(x+1)=42 equation:

Simplifying
3x(x + 1) = 42

Reorder the terms:
3x(1 + x) = 42
(1 * 3x + x * 3x) = 42
(3x + 3x2) = 42

Solving
3x + 3x2 = 42

Solving for variable 'x'.

Reorder the terms:
-42 + 3x + 3x2 = 42 + -42

Combine like terms: 42 + -42 = 0
-42 + 3x + 3x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-14 + x + x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-14 + x + x2)' equal to zero and attempt to solve:

Simplifying
-14 + x + x2 = 0

Solving
-14 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + x + 14 + x2 = 0 + 14

Reorder the terms:
-14 + 14 + x + x2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + x + x2 = 0 + 14
x + x2 = 0 + 14

Combine like terms: 0 + 14 = 14
x + x2 = 14

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 = 14 + 0.25

Reorder the terms:
0.25 + x + x2 = 14 + 0.25

Combine like terms: 14 + 0.25 = 14.25
0.25 + x + x2 = 14.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.774917218 and -3.774917218.

Subproblem 1
x + 0.5 = 3.774917218

Simplifying
x + 0.5 = 3.774917218

Reorder the terms:
0.5 + x = 3.774917218

Solving
0.5 + x = 3.774917218

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 = 3.774917218 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 3.774917218 + -0.5
x = 3.774917218 + -0.5

Combine like terms: 3.774917218 + -0.5 = 3.274917218
x = 3.274917218

Simplifying
x = 3.274917218

Subproblem 2
x + 0.5 = -3.774917218

Simplifying
x + 0.5 = -3.774917218

Reorder the terms:
0.5 + x = -3.774917218

Solving
0.5 + x = -3.774917218

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 = -3.774917218 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -3.774917218 + -0.5
x = -3.774917218 + -0.5

Combine like terms: -3.774917218 + -0.5 = -4.274917218
x = -4.274917218

Simplifying
x = -4.274917218

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.274917218, -4.274917218}
Solution
x = {3.274917218, -4.274917218}