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Complex Numbers And Quadratic Equations
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 the common root is
1
4
3
2
D.
2
Let α and 4β be roots of x^{2}– 6x + a = 0 and
α, 3β be the roots of x^{2}– cx + 6 = 0, then
α + 4β = 6 and 4αβ = a
α + 3β = c and 3αβ = 6.
We get αβ = 2 ⇒ a = 8
So the first equation is x^{2} – 6x + 8 = 0 ⇒ x = 2, 4
If α = 2 and 4β = 4 then 3β = 3
If α = 4 and 4β = 2, then 3β = 3/2 (noninteger)
∴ common root is x = 2.
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