3(x-3)+5(x-4)=(x-4)(x-3)

Solution for 3(x-3)+5(x-4)=(x-4)(x-3) equation:

Simplifying
3(x + -3) + 5(x + -4) = (x + -4)(x + -3)

Reorder the terms:
3(-3 + x) + 5(x + -4) = (x + -4)(x + -3)
(-3 * 3 + x * 3) + 5(x + -4) = (x + -4)(x + -3)
(-9 + 3x) + 5(x + -4) = (x + -4)(x + -3)

Reorder the terms:
-9 + 3x + 5(-4 + x) = (x + -4)(x + -3)
-9 + 3x + (-4 * 5 + x * 5) = (x + -4)(x + -3)
-9 + 3x + (-20 + 5x) = (x + -4)(x + -3)

Reorder the terms:
-9 + -20 + 3x + 5x = (x + -4)(x + -3)

Combine like terms: -9 + -20 = -29
-29 + 3x + 5x = (x + -4)(x + -3)

Combine like terms: 3x + 5x = 8x
-29 + 8x = (x + -4)(x + -3)

Reorder the terms:
-29 + 8x = (-4 + x)(x + -3)

Reorder the terms:
-29 + 8x = (-4 + x)(-3 + x)

Multiply (-4 + x) * (-3 + x)
-29 + 8x = (-4(-3 + x) + x(-3 + x))
-29 + 8x = ((-3 * -4 + x * -4) + x(-3 + x))
-29 + 8x = ((12 + -4x) + x(-3 + x))
-29 + 8x = (12 + -4x + (-3 * x + x * x))
-29 + 8x = (12 + -4x + (-3x + x2))

Combine like terms: -4x + -3x = -7x
-29 + 8x = (12 + -7x + x2)

Solving
-29 + 8x = 12 + -7x + x2

Solving for variable 'x'.

Reorder the terms:
-29 + -12 + 8x + 7x + -1x2 = 12 + -7x + x2 + -12 + 7x + -1x2

Combine like terms: -29 + -12 = -41
-41 + 8x + 7x + -1x2 = 12 + -7x + x2 + -12 + 7x + -1x2

Combine like terms: 8x + 7x = 15x
-41 + 15x + -1x2 = 12 + -7x + x2 + -12 + 7x + -1x2

Reorder the terms:
-41 + 15x + -1x2 = 12 + -12 + -7x + 7x + x2 + -1x2

Combine like terms: 12 + -12 = 0
-41 + 15x + -1x2 = 0 + -7x + 7x + x2 + -1x2
-41 + 15x + -1x2 = -7x + 7x + x2 + -1x2

Combine like terms: -7x + 7x = 0
-41 + 15x + -1x2 = 0 + x2 + -1x2
-41 + 15x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-41 + 15x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
41 + -15x + x2 = 0

Move the constant term to the right:

Add '-41' to each side of the equation.
41 + -15x + -41 + x2 = 0 + -41

Reorder the terms:
41 + -41 + -15x + x2 = 0 + -41

Combine like terms: 41 + -41 = 0
0 + -15x + x2 = 0 + -41
-15x + x2 = 0 + -41

Combine like terms: 0 + -41 = -41
-15x + x2 = -41

The x term is -15x.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15x + 56.25 + x2 = -41 + 56.25

Reorder the terms:
56.25 + -15x + x2 = -41 + 56.25

Combine like terms: -41 + 56.25 = 15.25
56.25 + -15x + x2 = 15.25

Factor a perfect square on the left side:
(x + -7.5)(x + -7.5) = 15.25

Calculate the square root of the right side: 3.905124838

Break this problem into two subproblems by setting 
(x + -7.5) equal to 3.905124838 and -3.905124838.

Subproblem 1
x + -7.5 = 3.905124838

Simplifying
x + -7.5 = 3.905124838

Reorder the terms:
-7.5 + x = 3.905124838

Solving
-7.5 + x = 3.905124838

Solving for variable 'x'.

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

Add '7.5' to each side of the equation.
-7.5 + 7.5 + x = 3.905124838 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + x = 3.905124838 + 7.5
x = 3.905124838 + 7.5

Combine like terms: 3.905124838 + 7.5 = 11.405124838
x = 11.405124838

Simplifying
x = 11.405124838

Subproblem 2
x + -7.5 = -3.905124838

Simplifying
x + -7.5 = -3.905124838

Reorder the terms:
-7.5 + x = -3.905124838

Solving
-7.5 + x = -3.905124838

Solving for variable 'x'.

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

Add '7.5' to each side of the equation.
-7.5 + 7.5 + x = -3.905124838 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + x = -3.905124838 + 7.5
x = -3.905124838 + 7.5

Combine like terms: -3.905124838 + 7.5 = 3.594875162
x = 3.594875162

Simplifying
x = 3.594875162

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {11.405124838, 3.594875162}