2(2x-5)=6x(5-x)

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Solution for 2(2x-5)=6x(5-x) equation: 


Simplifying
2(2x + -5) = 6x(5 + -1x)

Reorder the terms:
2(-5 + 2x) = 6x(5 + -1x)
(-5 * 2 + 2x * 2) = 6x(5 + -1x)
(-10 + 4x) = 6x(5 + -1x)
-10 + 4x = (5 * 6x + -1x * 6x)
-10 + 4x = (30x + -6x2)

Solving
-10 + 4x = 30x + -6x2

Solving for variable 'x'.

Combine like terms: 4x + -30x = -26x
-10 + -26x + 6x2 = 30x + -6x2 + -30x + 6x2

Reorder the terms:
-10 + -26x + 6x2 = 30x + -30x + -6x2 + 6x2

Combine like terms: 30x + -30x = 0
-10 + -26x + 6x2 = 0 + -6x2 + 6x2
-10 + -26x + 6x2 = -6x2 + 6x2

Combine like terms: -6x2 + 6x2 = 0
-10 + -26x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-5 + -13x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-5 + -13x + 3x2)' equal to zero and attempt to solve:

Simplifying
-5 + -13x + 3x2 = 0

Solving
-5 + -13x + 3x2 = 0

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

Divide each side by '3'.
-1.666666667 + -4.333333333x + x2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -4.333333333x + 1.666666667 + x2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -4.333333333x + x2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -4.333333333x + x2 = 0 + 1.666666667
-4.333333333x + x2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-4.333333333x + x2 = 1.666666667

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 = 1.666666667 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333x + x2 = 1.666666667 + 4.694444446

Combine like terms: 1.666666667 + 4.694444446 = 6.361111113
4.694444446 + -4.333333333x + x2 = 6.361111113

Factor a perfect square on the left side:
(x + -2.166666667)(x + -2.166666667) = 6.361111113

Calculate the square root of the right side: 2.522124325

Break this problem into two subproblems by setting 
(x + -2.166666667) equal to 2.522124325 and -2.522124325.


Subproblem 1
x + -2.166666667 = 2.522124325

Simplifying
x + -2.166666667 = 2.522124325

Reorder the terms:
-2.166666667 + x = 2.522124325

Solving
-2.166666667 + x = 2.522124325

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 = 2.522124325 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = 2.522124325 + 2.166666667
x = 2.522124325 + 2.166666667

Combine like terms: 2.522124325 + 2.166666667 = 4.688790992
x = 4.688790992

Simplifying
x = 4.688790992


Subproblem 2
x + -2.166666667 = -2.522124325

Simplifying
x + -2.166666667 = -2.522124325

Reorder the terms:
-2.166666667 + x = -2.522124325

Solving
-2.166666667 + x = -2.522124325

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 = -2.522124325 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = -2.522124325 + 2.166666667
x = -2.522124325 + 2.166666667

Combine like terms: -2.522124325 + 2.166666667 = -0.355457658
x = -0.355457658

Simplifying
x = -0.355457658


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.688790992, -0.355457658}
Solution
x = {4.688790992, -0.355457658}

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