Condition for common root(s)
Let ax2 + bx
+ c = 0 and dx2 + ex + f = 0 have a common root a(say).
Then.
aa2 + ba + c = 0 and
da2
+ ea + f = 0
Solving for a2 and a, we get 
i.e. a2 = 
and
a
= 
&⇒
(dc - af)2 = (bf - ce) (ae - bd)
which is the required
condition for the two equations to have a common root.
Note:
Condition for both the roots to be
common is 
Example 1
Find the conditions
on a, b, c, d such that equations 2ax3+bx2+cx +d =0 and
2ax2 + 3bx + 4c = 0 have a common root.
Solution:
Let ‘a’ be a
common root of the given two equations. .
Than 2aa3+ba2+ca +d = 0 …
(1)
and 2aa2 +3ba +4c = 0 …
(2)
Multiply (2) with a
and then subtract (1) from it, we get
2ba2 +3ca - d =0 …
(3)
Now (2) and (3) are quadratic having
a common root a, so
a2 =
, a
= -
Eliminating a
from these two equations we get,
(4bc + ad)2 =
(bd
+4c2 )( b2-ac) which is the required condition.
