Question

Find all real values of a for which the equation x 4 − 2ax2 + x + a 2 − a = 0 has all its roots real.

Answer 1

            HERE IS YOUR ANSWER

Step-by-step explanation:

Write p(x)=x4−2ax2+x+(a2−a) as :

p(x)=(−x2+x+(a−1))(−x2−x+a)

For this equation to have 4 non-real roots, we need the discriminants of both quadratic polynomials to be negative.

The discriminants of the quadratic factors of p(x) are :

D1=1+4(a−1)=4a−3 and

D2=1+4a.

Now, D1<0 and D2<0⟹a<−14.

So, for a<−14, all roots of p(x) are non-real.

Therefore, p(x) has real roots if and only if a≥−14.