3x(x+1)41=7+5

Solution for 3x(x+1)41=7+5 equation:

Simplifying
3x(x + 1) * 41 = 7 + 5

Reorder the terms:
3x(1 + x) * 41 = 7 + 5

Reorder the terms for easier multiplication:
3 * 41x(1 + x) = 7 + 5

Multiply 3 * 41
123x(1 + x) = 7 + 5
(1 * 123x + x * 123x) = 7 + 5
(123x + 123x2) = 7 + 5

Combine like terms: 7 + 5 = 12
123x + 123x2 = 12

Solving
123x + 123x2 = 12

Solving for variable 'x'.

Reorder the terms:
-12 + 123x + 123x2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + 123x + 123x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-4 + 41x + 41x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-4 + 41x + 41x2)' equal to zero and attempt to solve:

Simplifying
-4 + 41x + 41x2 = 0

Solving
-4 + 41x + 41x2 = 0

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

Divide each side by '41'.
-0.09756097561 + x + x2 = 0

Move the constant term to the right:

Add '0.09756097561' to each side of the equation.
-0.09756097561 + x + 0.09756097561 + x2 = 0 + 0.09756097561

Reorder the terms:
-0.09756097561 + 0.09756097561 + x + x2 = 0 + 0.09756097561

Combine like terms: -0.09756097561 + 0.09756097561 = 0.00000000000
0.00000000000 + x + x2 = 0 + 0.09756097561
x + x2 = 0 + 0.09756097561

Combine like terms: 0 + 0.09756097561 = 0.09756097561
x + x2 = 0.09756097561

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.
 + 0.25 + x2 = 0.09756097561 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 0.09756097561 + 0.25

Combine like terms: 0.09756097561 + 0.25 = 0.34756097561
1.25 + x2 = 0.34756097561

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.34756097561

Calculate the square root of the right side: 0.589543023

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.589543023 and -0.589543023.

Subproblem 1
x + 0.5 = 0.589543023

Simplifying
x + 0.5 = 0.589543023

Reorder the terms:
0.5 + x = 0.589543023

Solving
0.5 + x = 0.589543023

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 = 0.589543023 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.589543023 + -0.5
x = 0.589543023 + -0.5

Combine like terms: 0.589543023 + -0.5 = 0.089543023
x = 0.089543023

Simplifying
x = 0.089543023

Subproblem 2
x + 0.5 = -0.589543023

Simplifying
x + 0.5 = -0.589543023

Reorder the terms:
0.5 + x = -0.589543023

Solving
0.5 + x = -0.589543023

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 = -0.589543023 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.589543023 + -0.5
x = -0.589543023 + -0.5

Combine like terms: -0.589543023 + -0.5 = -1.089543023
x = -1.089543023

Simplifying
x = -1.089543023

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.089543023, -1.089543023}
Solution
x = {0.089543023, -1.089543023}