Solve for r:
20 r^3 - 20 r + 7.5 = 0
20 r^3 - 20 r + 7.5 = 20 r^3 - 20 r + 15/2:
20 r^3 - 20 r + 15/2 = 0
The left hand side factors into a product with four terms:
5/2 (2 r - 1) (4 r^2 + 2 r - 3) = 0
Multiply both sides by 2/5:
(2 r - 1) (4 r^2 + 2 r - 3) = 0
Split into two equations:
2 r - 1 = 0 or 4 r^2 + 2 r - 3 = 0
Add 1 to both sides:
2 r = 1 or 4 r^2 + 2 r - 3 = 0
Divide both sides by 2:
r = 1/2 or 4 r^2 + 2 r - 3 = 0
Divide both sides by 4:
r = 1/2 or r^2 + r/2 - 3/4 = 0
Add 3/4 to both sides:
r = 1/2 or r^2 + r/2 = 3/4
Add 1/16 to both sides:
r = 1/2 or r^2 + r/2 + 1/16 = 13/16
Write the left hand side as a square:
r = 1/2 or (r + 1/4)^2 = 13/16
Take the square root of both sides:
r = 1/2 or r + 1/4 = sqrt(13)/4 or r + 1/4 = -sqrt(13)/4
Subtract 1/4 from both sides:
r = 1/2 or r = sqrt(13)/4 - 1/4 or r + 1/4 = -sqrt(13)/4
Subtract 1/4 from both sides:
Answer: |r = 1/2 or r = sqrt(13)/4 - 1/4 or r = -1/4 - sqrt(13)/4
