Question

If x^2+bx+c=0 and bx^2+cx+a=0 have a common root and a,b,c are non zero real no.,then (a^3+b^3+c^3)/abc is ??

Answer 1

ans.3

Answer 2

have u tried it yet?

Answer 3

yes

Answer 4

well show ur work

Answer 5

well i reached upto factoring a^3+b^3+c^3.
and i found that putting sum of roots and product of root, there will be no numeral ans.

Answer 6

use this :

\( a^3+b^3+c^3=3abc+(a+b+c)(a^2+b^2+c^2-ab-ac-bc) \)

Answer 7

i used it

Answer 8

so we must try to show that
\( (a+b+c)(a^2+b^2+c^2-ab-ac-bc)=0 \)

Answer 9

am i right?

Answer 10

how will that work?

Answer 11

(a^3+b^3+c^3)/abc=3+(a+b+c)(a^2+b^2+c^2-ab-ac-bc)
---------------------------------
if =0
u have the answer =3

Answer 12

ok

Answer 13

i think the question is :

If \(ax^2+bx+c=0 \) and \(bx^2+cx+a=0 \) have a common root and a,b,c are non zero real no., then \( \large\frac{a^3+b^3+c^3}{abc} \) is ??

Answer 14

Condition for exactly one common root of \( ax^2 + bx + c =  0 \)  & \( Ax^2 + Bx + C =  0 \) is

\( \large (cA-aC)^2=(bC-cB)(aB-bA) \)

Answer 15

so for our case it becomes

\( \large (bc – a^2)^2 = (ab – c^2) (ac – b^2) \)


By expansion:

\( \large b^2c^2 + a^4 – 2a^2bc = a^2bc – ab^3 – ac^3 + b^2c^2 \) or

\( \large a(a^3 + b^3 + c^3) = 3a^2bc \) and \( \large a^3 + b^3 + c^3= 3abc \)