Question

Find real roots of 4/x² + 1 = 4/x​

Answer 1

Answer:

The equation x4+x2+1 is what we call a biquadratic equation since only even powers of x are involved.

Let y:=x2, then x4+x2+1=0 is equivalent to y2+y+1=0. Now, it is derived from quadratic formula that y2+y+1=(y−w)(y−w2). Recalling what is y, one has:

x4+x2+1=(x2−w)(x2−w2)=(x2−w)(x−w)(x+w).

The factorization is still not complete, x2−w can still be split into a product of two linear factors.

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