5x(x+3)+5=2x(x+1)

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Solution for 5x(x+3)+5=2x(x+1) equation: 


Simplifying
5x(x + 3) + 5 = 2x(x + 1)

Reorder the terms:
5x(3 + x) + 5 = 2x(x + 1)
(3 * 5x + x * 5x) + 5 = 2x(x + 1)
(15x + 5x2) + 5 = 2x(x + 1)

Reorder the terms:
5 + 15x + 5x2 = 2x(x + 1)

Reorder the terms:
5 + 15x + 5x2 = 2x(1 + x)
5 + 15x + 5x2 = (1 * 2x + x * 2x)
5 + 15x + 5x2 = (2x + 2x2)

Solving
5 + 15x + 5x2 = 2x + 2x2

Solving for variable 'x'.

Reorder the terms:
5 + 15x + -2x + 5x2 + -2x2 = 2x + 2x2 + -2x + -2x2

Combine like terms: 15x + -2x = 13x
5 + 13x + 5x2 + -2x2 = 2x + 2x2 + -2x + -2x2

Combine like terms: 5x2 + -2x2 = 3x2
5 + 13x + 3x2 = 2x + 2x2 + -2x + -2x2

Reorder the terms:
5 + 13x + 3x2 = 2x + -2x + 2x2 + -2x2

Combine like terms: 2x + -2x = 0
5 + 13x + 3x2 = 0 + 2x2 + -2x2
5 + 13x + 3x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
5 + 13x + 3x2 = 0

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

Divide each side by '3'.
1.666666667 + 4.333333333x + x2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + 4.333333333x + -1.666666667 + x2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + 4.333333333x + x2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + 4.333333333x + x2 = 0 + -1.666666667
4.333333333x + x2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
4.333333333x + x2 = -1.666666667

The x term is 4.333333333x.  Take half its coefficient (2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
4.333333333x + 4.694444446 + x2 = -1.666666667 + 4.694444446

Reorder the terms:
4.694444446 + 4.333333333x + x2 = -1.666666667 + 4.694444446

Combine like terms: -1.666666667 + 4.694444446 = 3.027777779
4.694444446 + 4.333333333x + x2 = 3.027777779

Factor a perfect square on the left side:
(x + 2.166666667)(x + 2.166666667) = 3.027777779

Calculate the square root of the right side: 1.740051085

Break this problem into two subproblems by setting 
(x + 2.166666667) equal to 1.740051085 and -1.740051085.


Subproblem 1
x + 2.166666667 = 1.740051085

Simplifying
x + 2.166666667 = 1.740051085

Reorder the terms:
2.166666667 + x = 1.740051085

Solving
2.166666667 + x = 1.740051085

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-2.166666667' to each side of the equation.
2.166666667 + -2.166666667 + x = 1.740051085 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + x = 1.740051085 + -2.166666667
x = 1.740051085 + -2.166666667

Combine like terms: 1.740051085 + -2.166666667 = -0.426615582
x = -0.426615582

Simplifying
x = -0.426615582


Subproblem 2
x + 2.166666667 = -1.740051085

Simplifying
x + 2.166666667 = -1.740051085

Reorder the terms:
2.166666667 + x = -1.740051085

Solving
2.166666667 + x = -1.740051085

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-2.166666667' to each side of the equation.
2.166666667 + -2.166666667 + x = -1.740051085 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + x = -1.740051085 + -2.166666667
x = -1.740051085 + -2.166666667

Combine like terms: -1.740051085 + -2.166666667 = -3.906717752
x = -3.906717752

Simplifying
x = -3.906717752


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.426615582, -3.906717752}

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