4x² - 4a²x + (a⁴ - b⁴) = 0

Using the splitting middle term - the middle term of the general equation is divided in two such values that:

Product = a.c

For the given equation a = 4 ;b = - 4a² ; c = (a⁴ - b⁴)

= 4. (a⁴ - b⁴)

= 4a⁴ - 4b⁴

And either of their sum or difference = b

= - 4a²

Thus the two terms are - 2(a² + b²) and - 2(a² - b²)

Difference = - 2(a² + b²) - 2(a² - b²)

= - 2a² - 2b² - 2a² + 2b²

= - 4a²

Product = - 2(a² + b²). - 2(a² - b²)

= 4(a² + b²)(a² - b²)

= 4. (a⁴ - b⁴)

(∵ using a² - b² = (a + b) (a - b))

⇒ 4x² - 4a²x + (a⁴ - b⁴) = 0

⇒ 4x² - 4a²x + ((a²)² – (b²)²) = 0

(∵ using a² - b² = (a + b) (a - b))

⇒ 4x² - 2(a² + b²) x - 2(a² - b²) x + (a² + b²) (a² - b²) = 0

⇒ 2x [2x - (a² + b²)] - (a² - b²) [2x - (a² + b²)] = 0

⇒ [2x - (a² + b²)] [2x - (a² - b²)] = 0

⇒ [2x - (a² + b²)] = 0 or [2x - (a² - b²)] = 0

Hence the roots of given equation are