3x(x+1)=22

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Solution for 3x(x+1)=22 equation: 


Simplifying
3x(x + 1) = 22

Reorder the terms:
3x(1 + x) = 22
(1 * 3x + x * 3x) = 22
(3x + 3x2) = 22

Solving
3x + 3x2 = 22

Solving for variable 'x'.

Reorder the terms:
-22 + 3x + 3x2 = 22 + -22

Combine like terms: 22 + -22 = 0
-22 + 3x + 3x2 = 0

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

Divide each side by '3'.
-7.333333333 + x + x2 = 0

Move the constant term to the right:

Add '7.333333333' to each side of the equation.
-7.333333333 + x + 7.333333333 + x2 = 0 + 7.333333333

Reorder the terms:
-7.333333333 + 7.333333333 + x + x2 = 0 + 7.333333333

Combine like terms: -7.333333333 + 7.333333333 = 0.000000000
0.000000000 + x + x2 = 0 + 7.333333333
x + x2 = 0 + 7.333333333

Combine like terms: 0 + 7.333333333 = 7.333333333
x + x2 = 7.333333333

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 = 7.333333333 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 7.333333333 + 0.25

Combine like terms: 7.333333333 + 0.25 = 7.583333333
1.25 + x2 = 7.583333333

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 7.583333333

Calculate the square root of the right side: 2.753785274

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.753785274 and -2.753785274.


Subproblem 1
x + 0.5 = 2.753785274

Simplifying
x + 0.5 = 2.753785274

Reorder the terms:
0.5 + x = 2.753785274

Solving
0.5 + x = 2.753785274

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 = 2.753785274 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.753785274 + -0.5
x = 2.753785274 + -0.5

Combine like terms: 2.753785274 + -0.5 = 2.253785274
x = 2.253785274

Simplifying
x = 2.253785274


Subproblem 2
x + 0.5 = -2.753785274

Simplifying
x + 0.5 = -2.753785274

Reorder the terms:
0.5 + x = -2.753785274

Solving
0.5 + x = -2.753785274

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 = -2.753785274 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.753785274 + -0.5
x = -2.753785274 + -0.5

Combine like terms: -2.753785274 + -0.5 = -3.253785274
x = -3.253785274

Simplifying
x = -3.253785274


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.253785274, -3.253785274}

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