r^2-3r-5=0

Solution for r^2-3r-5=0 equation:

Simplifying
r2 + -3r + -5 = 0

Reorder the terms:
-5 + -3r + r2 = 0

Solving
-5 + -3r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -3r + 5 + r2 = 0 + 5

Reorder the terms:
-5 + 5 + -3r + r2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -3r + r2 = 0 + 5
-3r + r2 = 0 + 5

Combine like terms: 0 + 5 = 5
-3r + r2 = 5

The r term is -3r.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3r + 2.25 + r2 = 5 + 2.25

Reorder the terms:
2.25 + -3r + r2 = 5 + 2.25

Combine like terms: 5 + 2.25 = 7.25
2.25 + -3r + r2 = 7.25

Factor a perfect square on the left side:
(r + -1.5)(r + -1.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(r + -1.5) equal to 2.692582404 and -2.692582404.

Subproblem 1
r + -1.5 = 2.692582404

Simplifying
r + -1.5 = 2.692582404

Reorder the terms:
-1.5 + r = 2.692582404

Solving
-1.5 + r = 2.692582404

Solving for variable 'r'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + r = 2.692582404 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + r = 2.692582404 + 1.5
r = 2.692582404 + 1.5

Combine like terms: 2.692582404 + 1.5 = 4.192582404
r = 4.192582404

Simplifying
r = 4.192582404

Subproblem 2
r + -1.5 = -2.692582404

Simplifying
r + -1.5 = -2.692582404

Reorder the terms:
-1.5 + r = -2.692582404

Solving
-1.5 + r = -2.692582404

Solving for variable 'r'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + r = -2.692582404 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + r = -2.692582404 + 1.5
r = -2.692582404 + 1.5

Combine like terms: -2.692582404 + 1.5 = -1.192582404
r = -1.192582404

Simplifying
r = -1.192582404

Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {4.192582404, -1.192582404}