10x(x-3)=5(3x-8)

Solution for 10x(x-3)=5(3x-8) equation:

Simplifying
10x(x + -3) = 5(3x + -8)

Reorder the terms:
10x(-3 + x) = 5(3x + -8)
(-3 * 10x + x * 10x) = 5(3x + -8)
(-30x + 10x2) = 5(3x + -8)

Reorder the terms:
-30x + 10x2 = 5(-8 + 3x)
-30x + 10x2 = (-8 * 5 + 3x * 5)
-30x + 10x2 = (-40 + 15x)

Solving
-30x + 10x2 = -40 + 15x

Solving for variable 'x'.

Reorder the terms:
40 + -30x + -15x + 10x2 = -40 + 15x + 40 + -15x

Combine like terms: -30x + -15x = -45x
40 + -45x + 10x2 = -40 + 15x + 40 + -15x

Reorder the terms:
40 + -45x + 10x2 = -40 + 40 + 15x + -15x

Combine like terms: -40 + 40 = 0
40 + -45x + 10x2 = 0 + 15x + -15x
40 + -45x + 10x2 = 15x + -15x

Combine like terms: 15x + -15x = 0
40 + -45x + 10x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(8 + -9x + 2x2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor '(8 + -9x + 2x2)' equal to zero and attempt to solve:

Simplifying
8 + -9x + 2x2 = 0

Solving
8 + -9x + 2x2 = 0

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

Divide each side by '2'.
4 + -4.5x + x2 = 0

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + -4.5x + -4 + x2 = 0 + -4

Reorder the terms:
4 + -4 + -4.5x + x2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -4.5x + x2 = 0 + -4
-4.5x + x2 = 0 + -4

Combine like terms: 0 + -4 = -4
-4.5x + x2 = -4

The x term is -4.5x.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5x + 5.0625 + x2 = -4 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = -4 + 5.0625

Combine like terms: -4 + 5.0625 = 1.0625
5.0625 + -4.5x + x2 = 1.0625

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(x + -2.25) equal to 1.030776406 and -1.030776406.

Subproblem 1
x + -2.25 = 1.030776406

Simplifying
x + -2.25 = 1.030776406

Reorder the terms:
-2.25 + x = 1.030776406

Solving
-2.25 + x = 1.030776406

Solving for variable 'x'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + x = 1.030776406 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + x = 1.030776406 + 2.25
x = 1.030776406 + 2.25

Combine like terms: 1.030776406 + 2.25 = 3.280776406
x = 3.280776406

Simplifying
x = 3.280776406

Subproblem 2
x + -2.25 = -1.030776406

Simplifying
x + -2.25 = -1.030776406

Reorder the terms:
-2.25 + x = -1.030776406

Solving
-2.25 + x = -1.030776406

Solving for variable 'x'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + x = -1.030776406 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + x = -1.030776406 + 2.25
x = -1.030776406 + 2.25

Combine like terms: -1.030776406 + 2.25 = 1.219223594
x = 1.219223594

Simplifying
x = 1.219223594

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.280776406, 1.219223594}
Solution
x = {3.280776406, 1.219223594}