65*(x-5)*x=180

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Solution for 65*(x-5)*x=180 equation: 


Simplifying
65(x + -5) * x = 180

Reorder the terms:
65(-5 + x) * x = 180

Reorder the terms for easier multiplication:
65x(-5 + x) = 180
(-5 * 65x + x * 65x) = 180
(-325x + 65x2) = 180

Solving
-325x + 65x2 = 180

Solving for variable 'x'.

Reorder the terms:
-180 + -325x + 65x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + -325x + 65x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-36 + -65x + 13x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-36 + -65x + 13x2)' equal to zero and attempt to solve:

Simplifying
-36 + -65x + 13x2 = 0

Solving
-36 + -65x + 13x2 = 0

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

Divide each side by '13'.
-2.769230769 + -5x + x2 = 0

Move the constant term to the right:

Add '2.769230769' to each side of the equation.
-2.769230769 + -5x + 2.769230769 + x2 = 0 + 2.769230769

Reorder the terms:
-2.769230769 + 2.769230769 + -5x + x2 = 0 + 2.769230769

Combine like terms: -2.769230769 + 2.769230769 = 0.000000000
0.000000000 + -5x + x2 = 0 + 2.769230769
-5x + x2 = 0 + 2.769230769

Combine like terms: 0 + 2.769230769 = 2.769230769
-5x + x2 = 2.769230769

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = 2.769230769 + 6.25

Reorder the terms:
6.25 + -5x + x2 = 2.769230769 + 6.25

Combine like terms: 2.769230769 + 6.25 = 9.019230769
6.25 + -5x + x2 = 9.019230769

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 9.019230769

Calculate the square root of the right side: 3.003203418

Break this problem into two subproblems by setting 
(x + -2.5) equal to 3.003203418 and -3.003203418.


Subproblem 1
x + -2.5 = 3.003203418

Simplifying
x + -2.5 = 3.003203418

Reorder the terms:
-2.5 + x = 3.003203418

Solving
-2.5 + x = 3.003203418

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = 3.003203418 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = 3.003203418 + 2.5
x = 3.003203418 + 2.5

Combine like terms: 3.003203418 + 2.5 = 5.503203418
x = 5.503203418

Simplifying
x = 5.503203418


Subproblem 2
x + -2.5 = -3.003203418

Simplifying
x + -2.5 = -3.003203418

Reorder the terms:
-2.5 + x = -3.003203418

Solving
-2.5 + x = -3.003203418

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = -3.003203418 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = -3.003203418 + 2.5
x = -3.003203418 + 2.5

Combine like terms: -3.003203418 + 2.5 = -0.503203418
x = -0.503203418

Simplifying
x = -0.503203418


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.503203418, -0.503203418}
Solution
x = {5.503203418, -0.503203418}

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