13. Consider the equation
$$
\sqrt{3 x^{2}-8 x+1}+\sqrt{9 x^{2}-24 x-8}=3 \text {. }
$$

It is known that the largest root of the equation is $-k$ 'times the smallest root. Find $k$.
13. Answer: 9 .

Let $y=\sqrt{3 x^{2}-8 x+1}$. Then the equation becomes
$$
y+\sqrt{3 y^{2}-11}=3 .
$$

Then $\sqrt{3 y^{2}-11}=3-y$. Squaring both sides, we have $3 y^{2}-11=9-6 y+y^{2}$; that is, $y^{2}+3 y-10=0$. Then $y=2$ or $y=-5$ (rejected because $y \geq 0$ ).
Solve $3 x^{2}-8 x+1=2^{2}$. Then $x=3$ and $x=-3^{-1}$. Hence, $k=3 / 3^{-1}=9$.