(x-2)(x-3)=(2x-5)(x+3)

Solution for (x-2)(x-3)=(2x-5)(x+3) equation:

Simplifying
(x + -2)(x + -3) = (2x + -5)(x + 3)

Reorder the terms:
(-2 + x)(x + -3) = (2x + -5)(x + 3)

Reorder the terms:
(-2 + x)(-3 + x) = (2x + -5)(x + 3)

Multiply (-2 + x) * (-3 + x)
(-2(-3 + x) + x(-3 + x)) = (2x + -5)(x + 3)
((-3 * -2 + x * -2) + x(-3 + x)) = (2x + -5)(x + 3)
((6 + -2x) + x(-3 + x)) = (2x + -5)(x + 3)
(6 + -2x + (-3 * x + x * x)) = (2x + -5)(x + 3)
(6 + -2x + (-3x + x2)) = (2x + -5)(x + 3)

Combine like terms: -2x + -3x = -5x
(6 + -5x + x2) = (2x + -5)(x + 3)

Reorder the terms:
6 + -5x + x2 = (-5 + 2x)(x + 3)

Reorder the terms:
6 + -5x + x2 = (-5 + 2x)(3 + x)

Multiply (-5 + 2x) * (3 + x)
6 + -5x + x2 = (-5(3 + x) + 2x * (3 + x))
6 + -5x + x2 = ((3 * -5 + x * -5) + 2x * (3 + x))
6 + -5x + x2 = ((-15 + -5x) + 2x * (3 + x))
6 + -5x + x2 = (-15 + -5x + (3 * 2x + x * 2x))
6 + -5x + x2 = (-15 + -5x + (6x + 2x2))

Combine like terms: -5x + 6x = 1x
6 + -5x + x2 = (-15 + 1x + 2x2)

Solving
6 + -5x + x2 = -15 + 1x + 2x2

Solving for variable 'x'.

Reorder the terms:
6 + 15 + -5x + -1x + x2 + -2x2 = -15 + 1x + 2x2 + 15 + -1x + -2x2

Combine like terms: 6 + 15 = 21
21 + -5x + -1x + x2 + -2x2 = -15 + 1x + 2x2 + 15 + -1x + -2x2

Combine like terms: -5x + -1x = -6x
21 + -6x + x2 + -2x2 = -15 + 1x + 2x2 + 15 + -1x + -2x2

Combine like terms: x2 + -2x2 = -1x2
21 + -6x + -1x2 = -15 + 1x + 2x2 + 15 + -1x + -2x2

Reorder the terms:
21 + -6x + -1x2 = -15 + 15 + 1x + -1x + 2x2 + -2x2

Combine like terms: -15 + 15 = 0
21 + -6x + -1x2 = 0 + 1x + -1x + 2x2 + -2x2
21 + -6x + -1x2 = 1x + -1x + 2x2 + -2x2

Combine like terms: 1x + -1x = 0
21 + -6x + -1x2 = 0 + 2x2 + -2x2
21 + -6x + -1x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
21 + -6x + -1x2 = 0

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

Divide each side by '-1'.
-21 + 6x + x2 = 0

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + 6x + 21 + x2 = 0 + 21

Reorder the terms:
-21 + 21 + 6x + x2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + 6x + x2 = 0 + 21
6x + x2 = 0 + 21

Combine like terms: 0 + 21 = 21
6x + x2 = 21

The x term is 6x.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6x + 9 + x2 = 21 + 9

Reorder the terms:
9 + 6x + x2 = 21 + 9

Combine like terms: 21 + 9 = 30
9 + 6x + x2 = 30

Factor a perfect square on the left side:
(x + 3)(x + 3) = 30

Calculate the square root of the right side: 5.477225575

Break this problem into two subproblems by setting 
(x + 3) equal to 5.477225575 and -5.477225575.

Subproblem 1
x + 3 = 5.477225575

Simplifying
x + 3 = 5.477225575

Reorder the terms:
3 + x = 5.477225575

Solving
3 + x = 5.477225575

Solving for variable 'x'.

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

Add '-3' to each side of the equation.
3 + -3 + x = 5.477225575 + -3

Combine like terms: 3 + -3 = 0
0 + x = 5.477225575 + -3
x = 5.477225575 + -3

Combine like terms: 5.477225575 + -3 = 2.477225575
x = 2.477225575

Simplifying
x = 2.477225575

Subproblem 2
x + 3 = -5.477225575

Simplifying
x + 3 = -5.477225575

Reorder the terms:
3 + x = -5.477225575

Solving
3 + x = -5.477225575

Solving for variable 'x'.

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

Add '-3' to each side of the equation.
3 + -3 + x = -5.477225575 + -3

Combine like terms: 3 + -3 = 0
0 + x = -5.477225575 + -3
x = -5.477225575 + -3

Combine like terms: -5.477225575 + -3 = -8.477225575
x = -8.477225575

Simplifying
x = -8.477225575

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.477225575, -8.477225575}