Question

Find the roots of the following equation, if they exist, by applying the quadratic formula:

12abx² – (9a² – 8b²)x – 6ab = 0, where a ≠ 0 and b ≠ 0                  

Answer 1

Given: 

12abx² – (9a² – 8b²)x – 6ab = 0

On comparing it with Ax² + Bx + C = 0, we get 

A = 12ab, B = –(9a² – 8b²) and C = – 6ab 

Discriminant D is given by:

D = B² – 4AC 

= [–(9a² – 8b²)]² – 4 × 12ab × (–6ab) 

= 81a⁴ – 144a²b² + 64b⁴ + 288a²b² 

= 81a⁴ + 144a²b² + 64b⁴ 

= (9a² + 8b²)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))/(2 xx 12ab)`

= `(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))/(2 xx 12ab)`

= `(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)`.