Solve the equation 4x² – 4a²x + (a⁴-b⁴) =0 by factorisation method.

Given: equation 4x² – 4a²x + (a⁴-b⁴) =0

To find: The roots by factorisation method

Method Used:

To solve the quadratic equation by factorisation method, follow the steps:

1) Multiply the coefficient of x² and constant term.

2) factorise the result obtained in step 1.

3) Now choose the pair of factors in such a way that after adding or subtracting(splitting) them

You get coefficient of x.

Explanation:

Here,

Coefficient of x² = 4

Constant term = (a⁴-b⁴)

As a²-b² = (a-b) (a +b)

(a⁴-b⁴) = (a² – b²) (a²+b²)

As 4(a⁴-b⁴) = 4[(a² – b²) (a²+b²)]

Coefficient of x = - 4a²

=- 2a² - 2a²

Add and subtract 2b² to get,

= -2a² - 2a² + 2b² - 2b²

= -2a² – 2b² – 2a² + 2b²

= - [2(a²+ b²) + 2(a²- b²)]

∴ 4x² – 4a²x + (a⁴-b⁴) = 0

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

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

⇒ [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 and 2x-(a²- b²) = 0

Hence roots are .