Two objects $A$ and $B$ are $343 \mathrm{~m}$ apart. Both start moving towards each other at the same moment with uniform acceleration. One object has an initial velocity of $3 \mathrm{~m}$ and an acceleration of $5 \mathrm{~m}$; the other object has an initial velocity of $4 \mathrm{~m}$ and an acceleration of $7 \mathrm{~m}$. The question is, after how much time will the two objects meet, and at what distance from their starting points?
If two bodies meet after $t \mathrm{~s}$, then the distance covered by $A$ is $3 t+\frac{5 t^{2}}{2}$, and the distance covered by $B$ is $4 t+\frac{7 t^{2}}{2}$; thus

$$
3 t+\frac{5 t^{2}}{2}+4 t+\frac{7 t^{2}}{2}=343
$$

or

$$
6 t^{2}+7 t=343
$$

from which

$$
t=\frac{-7 \pm 91}{12}
$$

Since in this case $t$ can only be positive, therefore $t=7 \mathrm{~s}$. During this time, $A$ covers $143.5 \mathrm{~m}$, and $B$ covers $199.5 \mathrm{~m}$.

Number of solutions: 27.