(x-1)(2x-3)=(x-2)(x-5)

Solution for (x-1)(2x-3)=(x-2)(x-5) equation:

Simplifying
(x + -1)(2x + -3) = (x + -2)(x + -5)

Reorder the terms:
(-1 + x)(2x + -3) = (x + -2)(x + -5)

Reorder the terms:
(-1 + x)(-3 + 2x) = (x + -2)(x + -5)

Multiply (-1 + x) * (-3 + 2x)
(-1(-3 + 2x) + x(-3 + 2x)) = (x + -2)(x + -5)
((-3 * -1 + 2x * -1) + x(-3 + 2x)) = (x + -2)(x + -5)
((3 + -2x) + x(-3 + 2x)) = (x + -2)(x + -5)
(3 + -2x + (-3 * x + 2x * x)) = (x + -2)(x + -5)
(3 + -2x + (-3x + 2x2)) = (x + -2)(x + -5)

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

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

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

Multiply (-2 + x) * (-5 + x)
3 + -5x + 2x2 = (-2(-5 + x) + x(-5 + x))
3 + -5x + 2x2 = ((-5 * -2 + x * -2) + x(-5 + x))
3 + -5x + 2x2 = ((10 + -2x) + x(-5 + x))
3 + -5x + 2x2 = (10 + -2x + (-5 * x + x * x))
3 + -5x + 2x2 = (10 + -2x + (-5x + x2))

Combine like terms: -2x + -5x = -7x
3 + -5x + 2x2 = (10 + -7x + x2)

Solving
3 + -5x + 2x2 = 10 + -7x + x2

Solving for variable 'x'.

Reorder the terms:
3 + -10 + -5x + 7x + 2x2 + -1x2 = 10 + -7x + x2 + -10 + 7x + -1x2

Combine like terms: 3 + -10 = -7
-7 + -5x + 7x + 2x2 + -1x2 = 10 + -7x + x2 + -10 + 7x + -1x2

Combine like terms: -5x + 7x = 2x
-7 + 2x + 2x2 + -1x2 = 10 + -7x + x2 + -10 + 7x + -1x2

Combine like terms: 2x2 + -1x2 = 1x2
-7 + 2x + 1x2 = 10 + -7x + x2 + -10 + 7x + -1x2

Reorder the terms:
-7 + 2x + 1x2 = 10 + -10 + -7x + 7x + x2 + -1x2

Combine like terms: 10 + -10 = 0
-7 + 2x + 1x2 = 0 + -7x + 7x + x2 + -1x2
-7 + 2x + 1x2 = -7x + 7x + x2 + -1x2

Combine like terms: -7x + 7x = 0
-7 + 2x + 1x2 = 0 + x2 + -1x2
-7 + 2x + 1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-7 + 2x + 1x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 2x + 7 + x2 = 0 + 7

Reorder the terms:
-7 + 7 + 2x + x2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 2x + x2 = 0 + 7
2x + x2 = 0 + 7

Combine like terms: 0 + 7 = 7
2x + x2 = 7

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 7 + 1

Reorder the terms:
1 + 2x + x2 = 7 + 1

Combine like terms: 7 + 1 = 8
1 + 2x + x2 = 8

Factor a perfect square on the left side:
(x + 1)(x + 1) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x + 1) equal to 2.828427125 and -2.828427125.

Subproblem 1
x + 1 = 2.828427125

Simplifying
x + 1 = 2.828427125

Reorder the terms:
1 + x = 2.828427125

Solving
1 + x = 2.828427125

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = 2.828427125 + -1

Combine like terms: 1 + -1 = 0
0 + x = 2.828427125 + -1
x = 2.828427125 + -1

Combine like terms: 2.828427125 + -1 = 1.828427125
x = 1.828427125

Simplifying
x = 1.828427125

Subproblem 2
x + 1 = -2.828427125

Simplifying
x + 1 = -2.828427125

Reorder the terms:
1 + x = -2.828427125

Solving
1 + x = -2.828427125

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = -2.828427125 + -1

Combine like terms: 1 + -1 = 0
0 + x = -2.828427125 + -1
x = -2.828427125 + -1

Combine like terms: -2.828427125 + -1 = -3.828427125
x = -3.828427125

Simplifying
x = -3.828427125

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.828427125, -3.828427125}