13.429 First, the motorboat traveled $a$ km across the lake, and then half of this distance up the river flowing into the lake. The entire trip lasted 1 hour. Find the own speed of the motorboat if the speed of the river current is $c$ km/h.
Solution. If $x$ is the speed of the motorboat (km/h), then $\frac{a}{x}$ is the time it takes to travel on the lake, and $\frac{a}{2(x-c)}$ is the time it takes to travel on the river.

From this,

$\frac{a}{x}+\frac{a}{2(x-c)}=1 \Rightarrow 2 x^{2}-x(2 c+3 a)+2 a c=0 \Rightarrow x=\frac{3 a+2 c \pm \sqrt{9 a^{2}-4 a c+4 c^{2}}}{4}$.

One root is discarded because $x>c$.

Answer: $\frac{3 a+2 c+\sqrt{9 a^{2}-4 a c+4 c^{2}}}{4} \text{km}/$ h.