3r(4r-5)=6(5r+8)

Solution for 3r(4r-5)=6(5r+8) equation:

Simplifying
3r(4r + -5) = 6(5r + 8)

Reorder the terms:
3r(-5 + 4r) = 6(5r + 8)
(-5 * 3r + 4r * 3r) = 6(5r + 8)
(-15r + 12r2) = 6(5r + 8)

Reorder the terms:
-15r + 12r2 = 6(8 + 5r)
-15r + 12r2 = (8 * 6 + 5r * 6)
-15r + 12r2 = (48 + 30r)

Solving
-15r + 12r2 = 48 + 30r

Solving for variable 'r'.

Reorder the terms:
-48 + -15r + -30r + 12r2 = 48 + 30r + -48 + -30r

Combine like terms: -15r + -30r = -45r
-48 + -45r + 12r2 = 48 + 30r + -48 + -30r

Reorder the terms:
-48 + -45r + 12r2 = 48 + -48 + 30r + -30r

Combine like terms: 48 + -48 = 0
-48 + -45r + 12r2 = 0 + 30r + -30r
-48 + -45r + 12r2 = 30r + -30r

Combine like terms: 30r + -30r = 0
-48 + -45r + 12r2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-16 + -15r + 4r2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-16 + -15r + 4r2)' equal to zero and attempt to solve:

Simplifying
-16 + -15r + 4r2 = 0

Solving
-16 + -15r + 4r2 = 0

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

Divide each side by '4'.
-4 + -3.75r + r2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -3.75r + 4 + r2 = 0 + 4

Reorder the terms:
-4 + 4 + -3.75r + r2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -3.75r + r2 = 0 + 4
-3.75r + r2 = 0 + 4

Combine like terms: 0 + 4 = 4
-3.75r + r2 = 4

The r term is -3.75r.  Take half its coefficient (-1.875).
Square it (3.515625) and add it to both sides.

Add '3.515625' to each side of the equation.
-3.75r + 3.515625 + r2 = 4 + 3.515625

Reorder the terms:
3.515625 + -3.75r + r2 = 4 + 3.515625

Combine like terms: 4 + 3.515625 = 7.515625
3.515625 + -3.75r + r2 = 7.515625

Factor a perfect square on the left side:
(r + -1.875)(r + -1.875) = 7.515625

Calculate the square root of the right side: 2.741464025

Break this problem into two subproblems by setting 
(r + -1.875) equal to 2.741464025 and -2.741464025.

Subproblem 1
r + -1.875 = 2.741464025

Simplifying
r + -1.875 = 2.741464025

Reorder the terms:
-1.875 + r = 2.741464025

Solving
-1.875 + r = 2.741464025

Solving for variable 'r'.

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

Add '1.875' to each side of the equation.
-1.875 + 1.875 + r = 2.741464025 + 1.875

Combine like terms: -1.875 + 1.875 = 0.000
0.000 + r = 2.741464025 + 1.875
r = 2.741464025 + 1.875

Combine like terms: 2.741464025 + 1.875 = 4.616464025
r = 4.616464025

Simplifying
r = 4.616464025

Subproblem 2
r + -1.875 = -2.741464025

Simplifying
r + -1.875 = -2.741464025

Reorder the terms:
-1.875 + r = -2.741464025

Solving
-1.875 + r = -2.741464025

Solving for variable 'r'.

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

Add '1.875' to each side of the equation.
-1.875 + 1.875 + r = -2.741464025 + 1.875

Combine like terms: -1.875 + 1.875 = 0.000
0.000 + r = -2.741464025 + 1.875
r = -2.741464025 + 1.875

Combine like terms: -2.741464025 + 1.875 = -0.866464025
r = -0.866464025

Simplifying
r = -0.866464025

Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {4.616464025, -0.866464025}
Solution
r = {4.616464025, -0.866464025}