Biquadratic equation

By Vieta's formula we find that the sum of roots = 4 and the product of roots = 1. So apply $A.M \geq G.M$ .

Let $\alpha$ represent the roots of the bi-quadratic equation .

$\frac { \sum_{i=1}^4 \alpha}{4} \geq ( \prod \alpha )^{ \frac{1}{4} }$

$\frac{4}{4} \geq 1$

Therefore , $A.M = G.M$ , which is possible only if all the roots of the biquadratic are equal . Now, all the roots are equal and we can easily find that each of the root equals 1 .

So the sum of squares of the roots ; $\sum_{i=1}^4 ( \alpha ^{2} ) =1+1+1+1 =4$