Find c+d if the roots of the cubic equation x^3-19(x^2)+cx+d=0 are three positive integers in a geometric sequence.

Question

There is an equation x³ + 19x² + cx + d = 0. There are three positive integer solutions to this equation, x₁, x₂, and x₃. It is known that x₁*r=x₂ and x₂*r=x₃. You are asked to find the value of c+d. 

Answer 1

Let the roots be a, ar and ar^2.

The the equation must be as follows:

(either multiply the factors or know the relation between roots and coefficidents):

x³ - a(1 + r + r²)x² + a²(1 + r + r²) - a³r³  and this must be identically equal to

 

x³ - 19x² + cx + d

 

This means a(1 + r + r²) = 19

 

From this you should be able to compute c + d in terms of a and r.

 

If you play around with the equation for the coefficient of the x² term you can show that one solution is a = 9 and r = 2/3, so the the roots are 9, 6 and 4.  There may be other solutions.