4x(3x+13)=180

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Solution for 4x(3x+13)=180 equation: 


Simplifying
4x(3x + 13) = 180

Reorder the terms:
4x(13 + 3x) = 180
(13 * 4x + 3x * 4x) = 180
(52x + 12x2) = 180

Solving
52x + 12x2 = 180

Solving for variable 'x'.

Reorder the terms:
-180 + 52x + 12x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 52x + 12x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-45 + 13x + 3x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-45 + 13x + 3x2)' equal to zero and attempt to solve:

Simplifying
-45 + 13x + 3x2 = 0

Solving
-45 + 13x + 3x2 = 0

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

Divide each side by '3'.
-15 + 4.333333333x + x2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 4.333333333x + 15 + x2 = 0 + 15

Reorder the terms:
-15 + 15 + 4.333333333x + x2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 4.333333333x + x2 = 0 + 15
4.333333333x + x2 = 0 + 15

Combine like terms: 0 + 15 = 15
4.333333333x + x2 = 15

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 = 15 + 4.694444446

Reorder the terms:
4.694444446 + 4.333333333x + x2 = 15 + 4.694444446

Combine like terms: 15 + 4.694444446 = 19.694444446
4.694444446 + 4.333333333x + x2 = 19.694444446

Factor a perfect square on the left side:
(x + 2.166666667)(x + 2.166666667) = 19.694444446

Calculate the square root of the right side: 4.437842319

Break this problem into two subproblems by setting 
(x + 2.166666667) equal to 4.437842319 and -4.437842319.


Subproblem 1
x + 2.166666667 = 4.437842319

Simplifying
x + 2.166666667 = 4.437842319

Reorder the terms:
2.166666667 + x = 4.437842319

Solving
2.166666667 + x = 4.437842319

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 = 4.437842319 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + x = 4.437842319 + -2.166666667
x = 4.437842319 + -2.166666667

Combine like terms: 4.437842319 + -2.166666667 = 2.271175652
x = 2.271175652

Simplifying
x = 2.271175652


Subproblem 2
x + 2.166666667 = -4.437842319

Simplifying
x + 2.166666667 = -4.437842319

Reorder the terms:
2.166666667 + x = -4.437842319

Solving
2.166666667 + x = -4.437842319

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 = -4.437842319 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + x = -4.437842319 + -2.166666667
x = -4.437842319 + -2.166666667

Combine like terms: -4.437842319 + -2.166666667 = -6.604508986
x = -6.604508986

Simplifying
x = -6.604508986


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.271175652, -6.604508986}
Solution
x = {2.271175652, -6.604508986}

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