(3x+12)(7x-32)=180

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Solution for (3x+12)(7x-32)=180 equation: 


Simplifying
(3x + 12)(7x + -32) = 180

Reorder the terms:
(12 + 3x)(7x + -32) = 180

Reorder the terms:
(12 + 3x)(-32 + 7x) = 180

Multiply (12 + 3x) * (-32 + 7x)
(12(-32 + 7x) + 3x * (-32 + 7x)) = 180
((-32 * 12 + 7x * 12) + 3x * (-32 + 7x)) = 180
((-384 + 84x) + 3x * (-32 + 7x)) = 180
(-384 + 84x + (-32 * 3x + 7x * 3x)) = 180
(-384 + 84x + (-96x + 21x2)) = 180

Combine like terms: 84x + -96x = -12x
(-384 + -12x + 21x2) = 180

Solving
-384 + -12x + 21x2 = 180

Solving for variable 'x'.

Reorder the terms:
-384 + -180 + -12x + 21x2 = 180 + -180

Combine like terms: -384 + -180 = -564
-564 + -12x + 21x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-564 + -12x + 21x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-188 + -4x + 7x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-188 + -4x + 7x2)' equal to zero and attempt to solve:

Simplifying
-188 + -4x + 7x2 = 0

Solving
-188 + -4x + 7x2 = 0

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

Divide each side by '7'.
-26.85714286 + -0.5714285714x + x2 = 0

Move the constant term to the right:

Add '26.85714286' to each side of the equation.
-26.85714286 + -0.5714285714x + 26.85714286 + x2 = 0 + 26.85714286

Reorder the terms:
-26.85714286 + 26.85714286 + -0.5714285714x + x2 = 0 + 26.85714286

Combine like terms: -26.85714286 + 26.85714286 = 0.00000000
0.00000000 + -0.5714285714x + x2 = 0 + 26.85714286
-0.5714285714x + x2 = 0 + 26.85714286

Combine like terms: 0 + 26.85714286 = 26.85714286
-0.5714285714x + x2 = 26.85714286

The x term is -0.5714285714x.  Take half its coefficient (-0.2857142857).
Square it (0.08163265305) and add it to both sides.

Add '0.08163265305' to each side of the equation.
-0.5714285714x + 0.08163265305 + x2 = 26.85714286 + 0.08163265305

Reorder the terms:
0.08163265305 + -0.5714285714x + x2 = 26.85714286 + 0.08163265305

Combine like terms: 26.85714286 + 0.08163265305 = 26.93877551305
0.08163265305 + -0.5714285714x + x2 = 26.93877551305

Factor a perfect square on the left side:
(x + -0.2857142857)(x + -0.2857142857) = 26.93877551305

Calculate the square root of the right side: 5.19025775

Break this problem into two subproblems by setting 
(x + -0.2857142857) equal to 5.19025775 and -5.19025775.


Subproblem 1
x + -0.2857142857 = 5.19025775

Simplifying
x + -0.2857142857 = 5.19025775

Reorder the terms:
-0.2857142857 + x = 5.19025775

Solving
-0.2857142857 + x = 5.19025775

Solving for variable 'x'.

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

Add '0.2857142857' to each side of the equation.
-0.2857142857 + 0.2857142857 + x = 5.19025775 + 0.2857142857

Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000
0.0000000000 + x = 5.19025775 + 0.2857142857
x = 5.19025775 + 0.2857142857

Combine like terms: 5.19025775 + 0.2857142857 = 5.4759720357
x = 5.4759720357

Simplifying
x = 5.4759720357


Subproblem 2
x + -0.2857142857 = -5.19025775

Simplifying
x + -0.2857142857 = -5.19025775

Reorder the terms:
-0.2857142857 + x = -5.19025775

Solving
-0.2857142857 + x = -5.19025775

Solving for variable 'x'.

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

Add '0.2857142857' to each side of the equation.
-0.2857142857 + 0.2857142857 + x = -5.19025775 + 0.2857142857

Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000
0.0000000000 + x = -5.19025775 + 0.2857142857
x = -5.19025775 + 0.2857142857

Combine like terms: -5.19025775 + 0.2857142857 = -4.9045434643
x = -4.9045434643

Simplifying
x = -4.9045434643


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.4759720357, -4.9045434643}
Solution
x = {5.4759720357, -4.9045434643}

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