Find the values of a, if ax2 – 4x + 9 = 0 has integral roots

For the roots to be integer the D that is the discriminat of the quadratic should be a perfect square and a= 1 with b and c should be integers which is howevr fulfilling the condtion.

Thereofre for a =1 the eqn. would have integera roots.

 

Let a=1/b, so the given equation becomes

x²-4bx+9b=0.This equation has integral roots 

if b is an integer. So discrimnant equals to 16b²-3b

and it must be a perfect square. Let k²=16b²-36b+81/4-81/4.So, k²=(4b-9/2) ²-81/4

After solving we will get: (8b-9-2k) (8b-9+2k) =1×81, 3×27, 9×9, 27×3, 81×1

It is satisfying only 81×1, 3×27.So b=3, -4. By this way we can conclude a=1/3, -1/4