Question

If α and β are the roots of ax^2 + 2bx + c = 0 where s (α+δ) and (β+δ) are the roots of Ax^2 + 2Bx + C = 0 for some constant δ, then prove that b^2 - ac / B^2 - AC = (a/A)^2 .

Answer 1

Answer:

Let α,β be roots of ax

3

+bx+c=0

γ,δ be roots of px

2

+2qx+r=0

β

α

=

δ

γ

…………(1)

and

α

β

=

γ

δ

………….(2)

(1)+(2)

⇒

β

α

+

α

β

=

δ

γ

+

γ

δ

=

αβ

2

2

+β

2

+2=

γδ

γ

2

+δ

2

+2

⇒

αβ

(α)

2

+(β)

2

+2αβ

=

γδ

γ

2

+δ

2

+2γδ

=

αβ

(α+β)

2

=

γδ

(γ+δ)

2

⇒

a

c

a

2

4b

2

=

p

r

p

2

4q

2

⇒

ac

b

2

=

p

r

q

2

.