The quadratic equations (x^2)-6x+a=0 and (x^2)-cx+6=0 have one root in common.The other two roots of the equations are integers and in ratio 3:4.Find the common root.

Dear Aparna,

 let common root is X and as other roots are in the ratio 3:4 so, let one root is x2 and other is x1 = 3x2/4

(x^2)-6x+a=0 :  Sum of roots = 6 = X + x2   ....1  and product = a = X*x2

(x^2)-cx+6=0 : Sum = c = X + 3x2/4         .....2 and product = 6 = X*x2/4

from 1 and 2 :     6-x2=c-3x2/4, so x2 = 24-4c, X = 24/24-4c      and a = 24

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