Question 77
The actual roots of the $n$ second degree equations are assumed to be known in the following table:
| Equation | Roots |
| :---: | :---: |
| $x^{2}+a_{1} x+b_{1}=0$ | $x_{0}, x_{1}$ |
| $x^{2}+a_{2} x+b_{2}=0$ | $x_{0}, x_{2}$ |
| $\cdots$ | $\cdots$ |
| $x^{2}+a_{n} x+b_{n}=0$ | $x_{0}, x_{n}$ |
Find, reasonably, the roots of the equation $$
x^{2}+\frac{a_{1}+a_{2}+\cdots+a_{n}}{n} x+\frac{b_{1}+b_{2}+\cdots+b_{n}}{n}=0
$$
| Equation | Roots |
| :---: | :---: |
| $x^{2}+a_{1} x+b_{1}=0$ | $x_{0}, x_{1}$ |
| $x^{2}+a_{2} x+b_{2}=0$ | $x_{0}, x_{2}$ |
| $\cdots$ | $\cdots$ |
| $x^{2}+a_{n} x+b_{n}=0$ | $x_{0}, x_{n}$ |
Find, reasonably, the roots of the equation $$
x^{2}+\frac{a_{1}+a_{2}+\cdots+a_{n}}{n} x+\frac{b_{1}+b_{2}+\cdots+b_{n}}{n}=0
$$