## Task 31/70

Determine all common solutions of the two equations

$$
\begin{array}{r}
3 x^{4}+13 x^{3}+20 x^{2}+17 x+7=0 \\
3 x^{4}+x^{3}-8 x^{2}+11 x-7=0
\end{array}
$$

without using an approximation method!
Subtracting (2) from (1), and then dividing by 2, we get

$$
6 x^{3}+14 x^{2}+3 x+7=0
$$

By adding the two equations, we obtain

$$
6 x^{4}+14 x^{3}+12 x^{2}+28 x=0
$$

Since $x=0$ is not a common solution, we can set $x \neq 0$ and divide (II) by $x$:

$$
6 x^{3}+14 x^{2}+12 x+28=0
$$

Subtracting (III) from (I) now yields

$$
-9 x-21=0 \quad \text { hence } \quad x=-\frac{7}{3}
$$

As the verification shows, this value is indeed a solution to both equations, and as the calculation demonstrates, it is also the only common solution.