9x(8x+3)=180

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Solution for 9x(8x+3)=180 equation: 


Simplifying
9x(8x + 3) = 180

Reorder the terms:
9x(3 + 8x) = 180
(3 * 9x + 8x * 9x) = 180
(27x + 72x2) = 180

Solving
27x + 72x2 = 180

Solving for variable 'x'.

Reorder the terms:
-180 + 27x + 72x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 27x + 72x2 = 0

Factor out the Greatest Common Factor (GCF), '9'.
9(-20 + 3x + 8x2) = 0

Ignore the factor 9.

Subproblem 1
Set the factor '(-20 + 3x + 8x2)' equal to zero and attempt to solve:

Simplifying
-20 + 3x + 8x2 = 0

Solving
-20 + 3x + 8x2 = 0

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

Divide each side by '8'.
-2.5 + 0.375x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + 0.375x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + 0.375x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + 0.375x + x2 = 0 + 2.5
0.375x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
0.375x + x2 = 2.5

The x term is 0.375x.  Take half its coefficient (0.1875).
Square it (0.03515625) and add it to both sides.

Add '0.03515625' to each side of the equation.
0.375x + 0.03515625 + x2 = 2.5 + 0.03515625

Reorder the terms:
0.03515625 + 0.375x + x2 = 2.5 + 0.03515625

Combine like terms: 2.5 + 0.03515625 = 2.53515625
0.03515625 + 0.375x + x2 = 2.53515625

Factor a perfect square on the left side:
(x + 0.1875)(x + 0.1875) = 2.53515625

Calculate the square root of the right side: 1.5922174

Break this problem into two subproblems by setting 
(x + 0.1875) equal to 1.5922174 and -1.5922174.


Subproblem 1
x + 0.1875 = 1.5922174

Simplifying
x + 0.1875 = 1.5922174

Reorder the terms:
0.1875 + x = 1.5922174

Solving
0.1875 + x = 1.5922174

Solving for variable 'x'.

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

Add '-0.1875' to each side of the equation.
0.1875 + -0.1875 + x = 1.5922174 + -0.1875

Combine like terms: 0.1875 + -0.1875 = 0.0000
0.0000 + x = 1.5922174 + -0.1875
x = 1.5922174 + -0.1875

Combine like terms: 1.5922174 + -0.1875 = 1.4047174
x = 1.4047174

Simplifying
x = 1.4047174


Subproblem 2
x + 0.1875 = -1.5922174

Simplifying
x + 0.1875 = -1.5922174

Reorder the terms:
0.1875 + x = -1.5922174

Solving
0.1875 + x = -1.5922174

Solving for variable 'x'.

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

Add '-0.1875' to each side of the equation.
0.1875 + -0.1875 + x = -1.5922174 + -0.1875

Combine like terms: 0.1875 + -0.1875 = 0.0000
0.0000 + x = -1.5922174 + -0.1875
x = -1.5922174 + -0.1875

Combine like terms: -1.5922174 + -0.1875 = -1.7797174
x = -1.7797174

Simplifying
x = -1.7797174


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.4047174, -1.7797174}
Solution
x = {1.4047174, -1.7797174}

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