3x(x+1)=16

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Solution for 3x(x+1)=16 equation: 


Simplifying
3x(x + 1) = 16

Reorder the terms:
3x(1 + x) = 16
(1 * 3x + x * 3x) = 16
(3x + 3x2) = 16

Solving
3x + 3x2 = 16

Solving for variable 'x'.

Reorder the terms:
-16 + 3x + 3x2 = 16 + -16

Combine like terms: 16 + -16 = 0
-16 + 3x + 3x2 = 0

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

Divide each side by '3'.
-5.333333333 + x + x2 = 0

Move the constant term to the right:

Add '5.333333333' to each side of the equation.
-5.333333333 + x + 5.333333333 + x2 = 0 + 5.333333333

Reorder the terms:
-5.333333333 + 5.333333333 + x + x2 = 0 + 5.333333333

Combine like terms: -5.333333333 + 5.333333333 = 0.000000000
0.000000000 + x + x2 = 0 + 5.333333333
x + x2 = 0 + 5.333333333

Combine like terms: 0 + 5.333333333 = 5.333333333
x + x2 = 5.333333333

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 = 5.333333333 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 5.333333333 + 0.25

Combine like terms: 5.333333333 + 0.25 = 5.583333333
1.25 + x2 = 5.583333333

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 5.583333333

Calculate the square root of the right side: 2.362907813

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.362907813 and -2.362907813.


Subproblem 1
x + 0.5 = 2.362907813

Simplifying
x + 0.5 = 2.362907813

Reorder the terms:
0.5 + x = 2.362907813

Solving
0.5 + x = 2.362907813

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 = 2.362907813 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.362907813 + -0.5
x = 2.362907813 + -0.5

Combine like terms: 2.362907813 + -0.5 = 1.862907813
x = 1.862907813

Simplifying
x = 1.862907813


Subproblem 2
x + 0.5 = -2.362907813

Simplifying
x + 0.5 = -2.362907813

Reorder the terms:
0.5 + x = -2.362907813

Solving
0.5 + x = -2.362907813

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 = -2.362907813 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.362907813 + -0.5
x = -2.362907813 + -0.5

Combine like terms: -2.362907813 + -0.5 = -2.862907813
x = -2.862907813

Simplifying
x = -2.862907813


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.862907813, -2.862907813}

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