condition for common roots

 Condition for common roots :

Let two quadratic equation:
  ax2 + bx + c = 0 & a’x2 + b’x + c’ = 0
(where  a.a’ ≠ 0  & ab’- a’b ≠ 0)
Let us assume a is  a common root , then
                ax2 + bx + c = 0 & a’x2 + b’x + c’ = 0
Put  x = a , then
             a a 2 + b a + c = 0  &
            a’ a 2 + b’ a + c’ = 0   
solve these two equation
using cross- multiplication
✪ Note :  To find common root between the two equation.
step1.  in both equation make the same coefficient of  x2 
step2. subtract equation one from another equation & find value of x , so this value of x is known as common root of these equation.
Example :  the equation x2 – x – 12 = 0 and  kx2 + 10x + 3 = 0 having one root is common in both equation. find value of k & also find common root.
solution : let a be a common root of these equation.
then a must be satisfies both equation    a2a – 12 = 0
and  ka2 + 10a + 3 = 0 
solving these two equations:
use above condition :
✪  ✪  ✪ ✪  ✪ 
= (bc’-b’c)/(ca’-c’a)
a= (ca’-c’a)/(ab’-a’b)
(bc’-b’c)(ab’-a’b)=(ca’-c’a)2
✪  ✪  ✪ ✪  ✪ 
(-3 + 120)( 10 + k) = (-12k-3)2
117(10+k)= 9(4k+1)2
(4k+1)2 = 13(10+k) 
16k2 + 8k + 1 = 130 +13k
16k2  – 5k – 129 = 0
16k2  – 48k + 43k – 129 = 0
k= 3 , or k = -43/16
a= ( -12k – 3)/(10 + k )
   = -3 or 4