Question

The quadratic equation x^{​​​​​​2}​​​​​-6x+a=0 and x^{​​​​​​2}​​​​​-cx+6=0 have one root in common. The other roots of the first and second equations are integers in 4:3, then find the common root.

Answer 1

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