sir this is quadratic equations based on common root problems

Let m be the common root. Then, 

 m^2+am+12=0,

m^2+bm+15=0, 

 

a = -(m^2+12)/m

b =  -(m^2+15)/m

 

Also, m^2+(a+b)m+15=0

 

Substituting the value of a and b 

 

m^2+(

 -(m^2+12)/m + 

 -(m^2+15)/m

)m+15=0

m^2 -2m^2-27+15 = 0

m^2 = -13

m = sqrt(13)i

 

a = -(m^2+12)/m = 1/sqrt(13)i = -sqrt(13)i/13

b =  -(m^2+15)/m = -2/sqrt(13)i = -2sqrt(13)i/13