The quadratic equations x 2 - 6 x + a = 0 and x 2 - c x + 6 = 0 have one root in common. The other roots of the first equation and the second equation are integers in the ratio 4 : 3 . Then the common root is

Step-by-step solution

Let, α be the common root and the other roots of the equations be 4 β and 3 β respectively. Then, α + 4 β = 6 , α 4 β = a α + 3 β = c , α 3 β = 6 ⇒ 4 3 = a 6 ⇒ α = 8 The first equation is x 2 - 6 x + 8 = 0 Whose roots are 2 and 4 If α = 2 ⇒ β = 1 So, roots of first equation is 2 , 4 and that of second equation is 2 , 3 If α = 4 ⇒ β = 1 2 ⇒ 3 β = 3 2 , 4 β = 2 Here the roots are not integers ⇒ α = 2