Problem 17. The side of a regular triangle $ABC$ is 4. Point $D$ is the midpoint of side $BC$. A line passing through $B$ intersects side $AC$ at point $M$. Perpendiculars from points $D$ and $A$ to line $BM$ are $DH$ and $AK$. Calculate the length of segment $AM$, if

$$
AK^4 - DH^4 = 15
$$
Instruction. Equation (1) has a unique solution. Let $\angle A B M=x$. Then $D H=2 \sin \left(60^{\circ}-x\right)$, $A K=4 \sin x$. Equation (1) takes the form:

$$
256 \sin ^{4} x-16 \sin ^{4}\left(60^{\circ}-x\right)=15
$$

Solving equation (2) on a microcalculator, we get $x=30^{\circ}$, and therefore, $A M=2$.