20. Find the sum of the fourth powers of the real roots of the equation

$$
x^{4}-1000 x^{2}+2017=0
$$
# Answer: 1991932

Solution: The roots of the equation are of the form $\pm \sqrt{t_{1}}, \pm \sqrt{t_{2}}$, where $t_{1,2}$ are the roots of the equation $t^{2}-1000 t+2017=0$. Therefore, the sum of the fourth powers is $2\left(t_{1}^{2}+t_{2}^{2}\right)=2\left(t_{1}+t_{2}\right)^{2}-4 t_{1} t_{2}=2 \cdot 1000^{2}-4 \cdot 2017=1991932$ - here we used Vieta's formulas.

## Variant 3-a