8x(x+9)=180

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


Simplifying
8x(x + 9) = 180

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

Solving
72x + 8x2 = 180

Solving for variable 'x'.

Reorder the terms:
-180 + 72x + 8x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 72x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-45 + 18x + 2x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-45 + 18x + 2x2)' equal to zero and attempt to solve:

Simplifying
-45 + 18x + 2x2 = 0

Solving
-45 + 18x + 2x2 = 0

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

Divide each side by '2'.
-22.5 + 9x + x2 = 0

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + 9x + 22.5 + x2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + 9x + x2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + 9x + x2 = 0 + 22.5
9x + x2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
9x + x2 = 22.5

The x term is 9x.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9x + 20.25 + x2 = 22.5 + 20.25

Reorder the terms:
20.25 + 9x + x2 = 22.5 + 20.25

Combine like terms: 22.5 + 20.25 = 42.75
20.25 + 9x + x2 = 42.75

Factor a perfect square on the left side:
(x + 4.5)(x + 4.5) = 42.75

Calculate the square root of the right side: 6.538348415

Break this problem into two subproblems by setting 
(x + 4.5) equal to 6.538348415 and -6.538348415.


Subproblem 1
x + 4.5 = 6.538348415

Simplifying
x + 4.5 = 6.538348415

Reorder the terms:
4.5 + x = 6.538348415

Solving
4.5 + x = 6.538348415

Solving for variable 'x'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + x = 6.538348415 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + x = 6.538348415 + -4.5
x = 6.538348415 + -4.5

Combine like terms: 6.538348415 + -4.5 = 2.038348415
x = 2.038348415

Simplifying
x = 2.038348415


Subproblem 2
x + 4.5 = -6.538348415

Simplifying
x + 4.5 = -6.538348415

Reorder the terms:
4.5 + x = -6.538348415

Solving
4.5 + x = -6.538348415

Solving for variable 'x'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + x = -6.538348415 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + x = -6.538348415 + -4.5
x = -6.538348415 + -4.5

Combine like terms: -6.538348415 + -4.5 = -11.038348415
x = -11.038348415

Simplifying
x = -11.038348415


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.038348415, -11.038348415}
Solution
x = {2.038348415, -11.038348415}

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