find three numbers in geometric progression whose sum is 19 and product is 216.
Ans:
Let the middle term be x and the common ratio be r. Then the 1st and 3rd terms are
x/r and x*r respectively.
Product = (x/r)*x*r*x = x^3 = 216.
So middle term x = 6.
Then the sum of the 3 terms = 6/r + 6 + 6*r = 19.

Multiplying by r
 This is a standard quadratic equation which can be
solved using the quadratic solver, as shown below.
The 2 roots are r = 2/3 and r = 3/2.
Hence the other 2 terms of the GP are (6*2/3) and (6/(2/3) = 4 and 9.
The 3 numbers are 4,6 and 9 (or 9,6 and 4).
Hope you got it :)
Solution using quadratic solver: 

 
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation  (in our case ) has the following solutons:
  
  
  
  For these solutions to exist, the discriminant  should not be a negative number.
  
  First, we need to compute the discriminant : .
  
  Discriminant d=25 is greater than zero. That means that there are two solutions: .
  
    
    
    
    Quadratic expression  can be factored:
  
  Again, the answer is: 1.5, 0.666666666666667.
Here's your graph: