The quadratic equations x² - 6x + a = 0 and x² - cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then what is the common root?

Let α and β are roots of: x² - 6x + a = 0
Let α and γ are roots of: x² - cx + 6 = 0
So, α + β = 6 ........ (i)
α.β = a ........ (ii)

α + γ = c ........ (iii)
α.γ = 6 ........ (iv)

β/γ = 4/3 ........ (v)

Now using (ii) and (iii)
α.β/α.γ = a/6
=> β/γ = a/6
=> using (v), 4/3 = a/6
=> So, a = 8

Now , x² - 6x + a = 0
x² - 6x + 8 = 0
x = 2 OR 6
As α.β = 8
So α = 2

Dear Student,
Please find below the solution to your problem.

Let α and β are roots of: x2 - 6x + a = 0
Let α and γ are roots of: x2 - cx + 6 = 0
So, α + β = 6 ........ (i)
α.β = a ........ (ii)
α + γ = c ........ (iii)
α.γ = 6 ........ (iv)
β/γ = 4/3 ........ (v)
Now using (ii) and (iii)
α.β/α.γ = a/6
=> β/γ = a/6
=> using (v), 4/3 = a/6
=> So, a = 8
Now , x2 - 6x + a = 0
x2 - 6x + 8 = 0
x = 2 OR 6
As α.β = 8
So α = 2

Thanks and Regards