Solve each of the following quadratic equations:

4x^{2} - 4a^{2}x + (a^{4} - b^{4}) = 0

4x^{2} - 4a^{2}x + (a^{4} - b^{4}) = 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 = - 4a^{2} ; c = (a^{4} - b^{4})

= 4. (a^{4} - b^{4})

= 4a^{4} - 4b^{4}

And either of their sum or difference = b

= - 4a^{2}

Thus the two terms are - 2(a^{2} + b^{2}) and - 2(a^{2} - b^{2})

Difference = - 2(a^{2} + b^{2}) - 2(a^{2} - b^{2})

= - 2a^{2} - 2b^{2} - 2a^{2} + 2b^{2}

= - 4a^{2}

Product = - 2(a^{2} + b^{2}). - 2(a^{2} - b^{2})

= 4(a^{2} + b^{2})(a^{2} - b^{2})

= 4. (a^{4} - b^{4})

(∵ using a^{2} - b^{2} = (a + b) (a - b))

⇒ 4x^{2} - 4a^{2}x + (a^{4} - b^{4}) = 0

⇒ 4x^{2} - 4a^{2}x + ((a^{2})^{2} – (b^{2})^{2}) = 0

(∵ using a^{2} - b^{2} = (a + b) (a - b))

⇒ 4x^{2} - 2(a^{2} + b^{2}) x - 2(a^{2} - b^{2}) x + (a^{2} + b^{2}) (a^{2} - b^{2}) = 0

⇒ 2x [2x - (a^{2} + b^{2})] - (a^{2} - b^{2}) [2x - (a^{2} + b^{2})] = 0

⇒ [2x - (a^{2} + b^{2})] [2x - (a^{2} - b^{2})] = 0

⇒ [2x - (a^{2} + b^{2})] = 0 or [2x - (a^{2} - b^{2})] = 0

Hence the roots of given equation are

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