(x-2)(x+3)-3=0

Solution for (x-2)(x+3)-3=0 equation:

Simplifying
(x + -2)(x + 3) + -3 = 0

Reorder the terms:
(-2 + x)(x + 3) + -3 = 0

Reorder the terms:
(-2 + x)(3 + x) + -3 = 0

Multiply (-2 + x) * (3 + x)
(-2(3 + x) + x(3 + x)) + -3 = 0
((3 * -2 + x * -2) + x(3 + x)) + -3 = 0
((-6 + -2x) + x(3 + x)) + -3 = 0
(-6 + -2x + (3 * x + x * x)) + -3 = 0
(-6 + -2x + (3x + x2)) + -3 = 0

Combine like terms: -2x + 3x = 1x
(-6 + 1x + x2) + -3 = 0

Reorder the terms:
-6 + -3 + 1x + x2 = 0

Combine like terms: -6 + -3 = -9
-9 + 1x + x2 = 0

Solving
-9 + 1x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 1x + 9 + x2 = 0 + 9

Reorder the terms:
-9 + 9 + 1x + x2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 1x + x2 = 0 + 9
1x + x2 = 0 + 9

Combine like terms: 0 + 9 = 9
1x + x2 = 9

The x term is 1x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 9 + 0.25

Reorder the terms:
0.25 + x + x2 = 9 + 0.25

Combine like terms: 9 + 0.25 = 9.25
0.25 + x + x2 = 9.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.041381265 and -3.041381265.

Subproblem 1
x + 0.5 = 3.041381265

Simplifying
x + 0.5 = 3.041381265

Reorder the terms:
0.5 + x = 3.041381265

Solving
0.5 + x = 3.041381265

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 3.041381265 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 3.041381265 + -0.5
x = 3.041381265 + -0.5

Combine like terms: 3.041381265 + -0.5 = 2.541381265
x = 2.541381265

Simplifying
x = 2.541381265

Subproblem 2
x + 0.5 = -3.041381265

Simplifying
x + 0.5 = -3.041381265

Reorder the terms:
0.5 + x = -3.041381265

Solving
0.5 + x = -3.041381265

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -3.041381265 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -3.041381265 + -0.5
x = -3.041381265 + -0.5

Combine like terms: -3.041381265 + -0.5 = -3.541381265
x = -3.541381265

Simplifying
x = -3.541381265

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.541381265, -3.541381265}