Quadratic Equation and Inequalities Part 1

We have, α is common root of the equations x2−x+2λ=0
and 3x2−10x+27λ=0
Now, common root of these equations is
(3α2−10α+27λ)−(3α2−3α+6λ)=0
⇒−7α+21λ=0
⇒α=3λ
Again, α is root of x2−x+2λ=0
∴α2−α+2λ=0
⇒(3λ)2−3λ+2λ=0
⇒9λ2−λ=0
⇒λ(9λ−1)=0
⇒λ=0,

1

9

⇒λ=

1

9

[∵ λ ≠ 0 ]
∴α=3λ=3×

1

9

=

1

3

Again, α and β are roots of the equation
x2−x+2λ=0
∴α+β=

−(−1)

1

=1
⇒β=1−α=1−

1

3

=

2

3

And α and γ are the roots of the equation 3x2−10x+27λ=0
∴α+γ=

−(−10)

3

=

10

3

⇒γ=

10

3

−α=

10

3

−

1

3

=

9

3

=3
∴

βγ

λ

=

( 2 3 )×(3)

( 1 9 )

=18