10x(4x+12)=180

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Solution for 10x(4x+12)=180 equation: 


Simplifying
10x(4x + 12) = 180

Reorder the terms:
10x(12 + 4x) = 180
(12 * 10x + 4x * 10x) = 180
(120x + 40x2) = 180

Solving
120x + 40x2 = 180

Solving for variable 'x'.

Reorder the terms:
-180 + 120x + 40x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 120x + 40x2 = 0

Factor out the Greatest Common Factor (GCF), '20'.
20(-9 + 6x + 2x2) = 0

Ignore the factor 20.

Subproblem 1
Set the factor '(-9 + 6x + 2x2)' equal to zero and attempt to solve:

Simplifying
-9 + 6x + 2x2 = 0

Solving
-9 + 6x + 2x2 = 0

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

Divide each side by '2'.
-4.5 + 3x + x2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 3x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 3x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 3x + x2 = 0 + 4.5
3x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
3x + x2 = 4.5

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 4.5 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 4.5 + 2.25

Combine like terms: 4.5 + 2.25 = 6.75
2.25 + 3x + x2 = 6.75

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 6.75

Calculate the square root of the right side: 2.598076211

Break this problem into two subproblems by setting 
(x + 1.5) equal to 2.598076211 and -2.598076211.


Subproblem 1
x + 1.5 = 2.598076211

Simplifying
x + 1.5 = 2.598076211

Reorder the terms:
1.5 + x = 2.598076211

Solving
1.5 + x = 2.598076211

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 2.598076211 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 2.598076211 + -1.5
x = 2.598076211 + -1.5

Combine like terms: 2.598076211 + -1.5 = 1.098076211
x = 1.098076211

Simplifying
x = 1.098076211


Subproblem 2
x + 1.5 = -2.598076211

Simplifying
x + 1.5 = -2.598076211

Reorder the terms:
1.5 + x = -2.598076211

Solving
1.5 + x = -2.598076211

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -2.598076211 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -2.598076211 + -1.5
x = -2.598076211 + -1.5

Combine like terms: -2.598076211 + -1.5 = -4.098076211
x = -4.098076211

Simplifying
x = -4.098076211


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.098076211, -4.098076211}
Solution
x = {1.098076211, -4.098076211}

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