(r+3)(r-5)=6

Solution for (r+3)(r-5)=6 equation:

Simplifying
(r + 3)(r + -5) = 6

Reorder the terms:
(3 + r)(r + -5) = 6

Reorder the terms:
(3 + r)(-5 + r) = 6

Multiply (3 + r) * (-5 + r)
(3(-5 + r) + r(-5 + r)) = 6
((-5 * 3 + r * 3) + r(-5 + r)) = 6
((-15 + 3r) + r(-5 + r)) = 6
(-15 + 3r + (-5 * r + r * r)) = 6
(-15 + 3r + (-5r + r2)) = 6

Combine like terms: 3r + -5r = -2r
(-15 + -2r + r2) = 6

Solving
-15 + -2r + r2 = 6

Solving for variable 'r'.

Reorder the terms:
-15 + -6 + -2r + r2 = 6 + -6

Combine like terms: -15 + -6 = -21
-21 + -2r + r2 = 6 + -6

Combine like terms: 6 + -6 = 0
-21 + -2r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + -2r + 21 + r2 = 0 + 21

Reorder the terms:
-21 + 21 + -2r + r2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + -2r + r2 = 0 + 21
-2r + r2 = 0 + 21

Combine like terms: 0 + 21 = 21
-2r + r2 = 21

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

Add '1' to each side of the equation.
-2r + 1 + r2 = 21 + 1

Reorder the terms:
1 + -2r + r2 = 21 + 1

Combine like terms: 21 + 1 = 22
1 + -2r + r2 = 22

Factor a perfect square on the left side:
(r + -1)(r + -1) = 22

Calculate the square root of the right side: 4.69041576

Break this problem into two subproblems by setting 
(r + -1) equal to 4.69041576 and -4.69041576.

Subproblem 1
r + -1 = 4.69041576

Simplifying
r + -1 = 4.69041576

Reorder the terms:
-1 + r = 4.69041576

Solving
-1 + r = 4.69041576

Solving for variable 'r'.

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

Add '1' to each side of the equation.
-1 + 1 + r = 4.69041576 + 1

Combine like terms: -1 + 1 = 0
0 + r = 4.69041576 + 1
r = 4.69041576 + 1

Combine like terms: 4.69041576 + 1 = 5.69041576
r = 5.69041576

Simplifying
r = 5.69041576

Subproblem 2
r + -1 = -4.69041576

Simplifying
r + -1 = -4.69041576

Reorder the terms:
-1 + r = -4.69041576

Solving
-1 + r = -4.69041576

Solving for variable 'r'.

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

Add '1' to each side of the equation.
-1 + 1 + r = -4.69041576 + 1

Combine like terms: -1 + 1 = 0
0 + r = -4.69041576 + 1
r = -4.69041576 + 1

Combine like terms: -4.69041576 + 1 = -3.69041576
r = -3.69041576

Simplifying
r = -3.69041576

Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {5.69041576, -3.69041576}