Solve for r:
sqrt(5 r - 9) - 3 = sqrt(r + 4) - 2
Add 3 to both sides:
sqrt(5 r - 9) = sqrt(r + 4) + 1
Raise both sides to the power of two:
5 r - 9 = (sqrt(r + 4) + 1)^2
(sqrt(r + 4) + 1)^2 = 5 + r + 2 sqrt(r + 4):
5 r - 9 = 5 + r + 2 sqrt(r + 4)
Subtract -9 + 5 r + 2 sqrt(r + 4) from both sides:
-2 sqrt(r + 4) = 14 - 4 r
Raise both sides to the power of two:
4 (r + 4) = (14 - 4 r)^2
Expand out terms of the left hand side:
4 r + 16 = (14 - 4 r)^2
Expand out terms of the right hand side:
4 r + 16 = 16 r^2 - 112 r + 196
Subtract 16 r^2 - 112 r + 196 from both sides:
-16 r^2 + 116 r - 180 = 0
The left hand side factors into a product with three terms:
-4 (r - 5) (4 r - 9) = 0
Divide both sides by -4:
(r - 5) (4 r - 9) = 0
Split into two equations:
r - 5 = 0 or 4 r - 9 = 0
Add 5 to both sides:
r = 5 or 4 r - 9 = 0
Add 9 to both sides:
r = 5 or 4 r = 9
Divide both sides by 4:
r = 5 or r = 9/4
sqrt(5 r - 9) - 3 ⇒ sqrt(5×9/4 - 9) - 3 = -3/2
sqrt(r + 4) - 2 ⇒ sqrt(9/4 + 4) - 2 = 1/2:
So this solution is incorrect
sqrt(5 r - 9) - 3 ⇒ sqrt(5×5 - 9) - 3 = 1
sqrt(r + 4) - 2 ⇒ sqrt(4 + 5) - 2 = 1:
So this solution is correct
The solution is:
r = 5
