The roots of the equation $(a^4+b^4)x^2+4abcd x + (c^4+d^4)$will be?

Question

The discriminant shall be $$16(abcd)^2-4a^4c^4-4a^4d^4-4b^4c^4-4b^4d^4$$ $$-4(b^2d^2+a^2c^2)^2-4(a^2d^2+b^2d^2)^2$$ which is clearly a negative value. So the roots should be imaginary. But the answer given is real and equal. How is that possible?

Answer 1

It will be $$-4(a^2b^2-c^2d^2)^2-4(a^2d^2-b^2c^2)^2\le0$$

Answer 2

The discriminant is given by $$-a^4c^4-a^4d^4+4a^2b^2c^2d^2-b^4c^4-b^4a^4$$ and now completing the squares.

Answer 3

Your discriminant is written slightly wrong it should be $$-4(b^2d^2 - a^2c^2)^2 -4(a^2d^2 - b^2c^2)^2.$$ Further as per the equation you have written you are surely correct that it will have imaginary roots. May be the answer given is wrong or some more information may be provided. Hope this helps.