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The quadratic equations 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 the ratio 4 : 3. Then what is the common root?

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?

Grade:12

2 Answers

ronit bhatiya
14 Points
13 years ago

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

Rishi Sharma
askIITians Faculty 646 Points
2 years ago
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

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