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

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

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

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

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

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

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

Solving
6 + -5x + x2 = 5

Solving for variable 'x'.

Reorder the terms:
6 + -5 + -5x + x2 = 5 + -5

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

Combine like terms: 5 + -5 = 0
1 + -5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -5x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -5x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -5x + x2 = 0 + -1
-5x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-5x + x2 = -1

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -1 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -1 + 6.25

Combine like terms: -1 + 6.25 = 5.25
6.25 + -5x + x2 = 5.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x + -2.5) equal to 2.291287847 and -2.291287847.

Subproblem 1
x + -2.5 = 2.291287847

Simplifying
x + -2.5 = 2.291287847

Reorder the terms:
-2.5 + x = 2.291287847

Solving
-2.5 + x = 2.291287847

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = 2.291287847 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = 2.291287847 + 2.5
x = 2.291287847 + 2.5

Combine like terms: 2.291287847 + 2.5 = 4.791287847
x = 4.791287847

Simplifying
x = 4.791287847

Subproblem 2
x + -2.5 = -2.291287847

Simplifying
x + -2.5 = -2.291287847

Reorder the terms:
-2.5 + x = -2.291287847

Solving
-2.5 + x = -2.291287847

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = -2.291287847 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = -2.291287847 + 2.5
x = -2.291287847 + 2.5

Combine like terms: -2.291287847 + 2.5 = 0.208712153
x = 0.208712153

Simplifying
x = 0.208712153

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.791287847, 0.208712153}