Question

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

a²b²x² – (4b⁴ – 3a⁴)x – 12a²b² = 0, a ≠ 0 and b ≠ 0

Answer 1

The given equation is a²b²x² – (4b⁴ – 3a⁴)x – 12a²b² = 0, a ≠ 0 and b ≠ 0

Comparing it with Ax² + Bx + C = 0, we get 

A = a²b², B = –(4b⁴ – 3a⁴) and C = –12a²b² 

∴ Discriminant, 

B² – 4AC = [–(4b⁴ – 3a⁴)]² – 4 × a²b² × (–12a²b²)

= 16b⁸ – 24a⁴b⁴ + 9a⁸ + 48a⁴b⁴ 

= 16b⁸ + 24a⁴b⁴ + 9a⁸

= (4b⁴ + 3a⁴)² > 0 

So, the given equation has real roots.

Now, `sqrt(D) = sqrt((4b^4 + 3a^4)^2) = 4b^4 + 3a^4` 

∴ `α = (-B + sqrt(D))/(2A)`

= `(-[-(4b^4 - 3a^4)] + (4b^4 + 3a^4))/(2 xx a^2b^2)`

= `(8b^4)/(2a^2b^2)`

= `(4b^2)/(a^2)` 

`β = (-B - sqrt(D))/(2A)`

= `(-[-(4b^4 - 3a^4)] + (4b^4 + 3a^4))/(2 xx a^2b^2)`

= `(-6a^4)/(2a^2b^2)`

= `(-3a^2)/(b^2)` 

Hence, `(4b^2)/(a^2)` and `(-3a^2)/(b^2)` are the roots of the given equation.