Question

`12abx^2-(9a^2-8b^2)x-6ab=0,` `Where  a≠0 and b≠0`                                                                                                                     

Answer 1

Given: 

`12abx^2-(9a^2-8b^2)x-6ab=0,` 

On comparing it with `Ax^2+Bx+C=0`, We get 

`A=12ab, B=-(9a^2-8b^2)x-6ab=0` 

Discriminant D is given by: 

`D=B^2-4AC` 

=`[-(9a^2-8b^2)]^2-4xx12abxx(-6ab)` 

=`81a^4-144a^2b^2+64b^4+288a^2b^2` 

=`81a^4+144a^2b^2+64b^2` 

=`(9a^2+8b^2)^2>0` 

Hence, the roots of the equation are equal.
Roots α and β are given by:  

α=`(-B+sqrt(D))/(2A)=-([-9a^2-8b^2]+(sqrt(9a^2+8b^2)^2))/(2xx12ab)=(9a^2-8b^2+9a^2+8b^2)/(24ab)=(18a^2)/(24ab)=(3a)/(4b)` 

`β=(-B-sqrt(D))/(2A)=-([-9a^2+8b^2]-(sqrt(9a^2-8b^2)^2))/(2xx12ab)=(9a^2-8b^2-9a^2-8b^2)/(24ab)=(-16a^2)/(24ab)=(-2b)/(3a)`

Thus, the roots of the equation are `(3a)/(4b)` and `(-2b)/(3a)`