5x(3x+2)=180

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Solution for 5x(3x+2)=180 equation: 


Simplifying
5x(3x + 2) = 180

Reorder the terms:
5x(2 + 3x) = 180
(2 * 5x + 3x * 5x) = 180
(10x + 15x2) = 180

Solving
10x + 15x2 = 180

Solving for variable 'x'.

Reorder the terms:
-180 + 10x + 15x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 10x + 15x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-36 + 2x + 3x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-36 + 2x + 3x2)' equal to zero and attempt to solve:

Simplifying
-36 + 2x + 3x2 = 0

Solving
-36 + 2x + 3x2 = 0

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

Divide each side by '3'.
-12 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 0.6666666667x + 12 + x2 = 0 + 12

Reorder the terms:
-12 + 12 + 0.6666666667x + x2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 0.6666666667x + x2 = 0 + 12
0.6666666667x + x2 = 0 + 12

Combine like terms: 0 + 12 = 12
0.6666666667x + x2 = 12

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 12 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 12 + 0.1111111112

Combine like terms: 12 + 0.1111111112 = 12.1111111112
0.1111111112 + 0.6666666667x + x2 = 12.1111111112

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 12.1111111112

Calculate the square root of the right side: 3.48010217

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 3.48010217 and -3.48010217.


Subproblem 1
x + 0.3333333334 = 3.48010217

Simplifying
x + 0.3333333334 = 3.48010217

Reorder the terms:
0.3333333334 + x = 3.48010217

Solving
0.3333333334 + x = 3.48010217

Solving for variable 'x'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = 3.48010217 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = 3.48010217 + -0.3333333334
x = 3.48010217 + -0.3333333334

Combine like terms: 3.48010217 + -0.3333333334 = 3.1467688366
x = 3.1467688366

Simplifying
x = 3.1467688366


Subproblem 2
x + 0.3333333334 = -3.48010217

Simplifying
x + 0.3333333334 = -3.48010217

Reorder the terms:
0.3333333334 + x = -3.48010217

Solving
0.3333333334 + x = -3.48010217

Solving for variable 'x'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = -3.48010217 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = -3.48010217 + -0.3333333334
x = -3.48010217 + -0.3333333334

Combine like terms: -3.48010217 + -0.3333333334 = -3.8134355034
x = -3.8134355034

Simplifying
x = -3.8134355034


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.1467688366, -3.8134355034}
Solution
x = {3.1467688366, -3.8134355034}

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