
A2. Find the largest possible value of the expression $\left|\sqrt{x^{2}+4 x+8}-\sqrt{x^{2}+8 x+17}\right|$ where $x$ is a real number.


Solution. We observe that

$$
\left|\sqrt{x^{2}+4 x+8}-\sqrt{x^{2}+8 x+17}\right|=\left|\sqrt{(x-(-2))^{2}+(0-2)^{2}}-\sqrt{\left.(x-(-4))^{2}+(0-1)^{2}\right)}\right|
$$

is the absolute difference of the distances from the point $P(x, 0)$ in the $x y$-plane to the points $A(-2,2)$ and $B(-4,1)$.

By the Triangle Inequality, $|P A-P B| \leq|A B|$ and the equality occurs exactly when $P$ lies on the line passing through $A$ and $B$, but not between them.

If $P, A, B$ are collinear, then $(x-(-2)) /(0-2)=((-4)-(-2)) /(1-2)$. This gives $x=-6$, and as $-6<-4<-2$,

$$
\left|\sqrt{(-6)^{2}+4(-6)+8}-\sqrt{(-6)^{2}+8(-6)+17}\right|=|\sqrt{20}-\sqrt{5}|=\sqrt{5}
$$

is the largest possible value of the expression.
