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

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

Given: 4x^{2} – 4a^{2}x + (a^{4} – b^{4}) = 0

Comparing with standard quadratic equation Ax^{2} + Bx + C = 0

A = 4, B = – 4a^{2}, C = (a^{4} – b^{4})

Discriminant D = B^{2} – 4AC

= (– 4a^{2})^{2} – 4.4. (a^{4} – b^{4})

= 16a^{4} + 16b^{4} – 16a^{4} = 16 b^{4} > 0

Hence the roots of equation are real.

Roots are given by

Hence the roots of equation are

31