2r^2-9r+6=0

Solution for 2r^2-9r+6=0 equation:

Simplifying
2r2 + -9r + 6 = 0

Reorder the terms:
6 + -9r + 2r2 = 0

Solving
6 + -9r + 2r2 = 0

Solving for variable 'r'.

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

Divide each side by '2'.
3 + -4.5r + r2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -4.5r + -3 + r2 = 0 + -3

Reorder the terms:
3 + -3 + -4.5r + r2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -4.5r + r2 = 0 + -3
-4.5r + r2 = 0 + -3

Combine like terms: 0 + -3 = -3
-4.5r + r2 = -3

The r term is -4.5r.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5r + 5.0625 + r2 = -3 + 5.0625

Reorder the terms:
5.0625 + -4.5r + r2 = -3 + 5.0625

Combine like terms: -3 + 5.0625 = 2.0625
5.0625 + -4.5r + r2 = 2.0625

Factor a perfect square on the left side:
(r + -2.25)(r + -2.25) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(r + -2.25) equal to 1.436140662 and -1.436140662.

Subproblem 1
r + -2.25 = 1.436140662

Simplifying
r + -2.25 = 1.436140662

Reorder the terms:
-2.25 + r = 1.436140662

Solving
-2.25 + r = 1.436140662

Solving for variable 'r'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + r = 1.436140662 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + r = 1.436140662 + 2.25
r = 1.436140662 + 2.25

Combine like terms: 1.436140662 + 2.25 = 3.686140662
r = 3.686140662

Simplifying
r = 3.686140662

Subproblem 2
r + -2.25 = -1.436140662

Simplifying
r + -2.25 = -1.436140662

Reorder the terms:
-2.25 + r = -1.436140662

Solving
-2.25 + r = -1.436140662

Solving for variable 'r'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + r = -1.436140662 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + r = -1.436140662 + 2.25
r = -1.436140662 + 2.25

Combine like terms: -1.436140662 + 2.25 = 0.813859338
r = 0.813859338

Simplifying
r = 0.813859338

Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {3.686140662, 0.813859338}