Solving 2-degree equations in 3 variables. We are given 3 equations:
$x^2+\sqrt3 xy + y^2 = 25$
$y^2 + z^2 = 9$
$x^2 +xz+ z^2 = 16$.
$x,y,z$ are positive real numbers.
Then we have to find value of $xy + 2yz + \sqrt3 xz$.

Note that you have $$\forall x,y,z \in \mathbb{R}^{+}: \left\{\begin{aligned} x^{2}+\sqrt{3} xy + y^2 &=& 25\\
y^{2} + z^{2} &=& 9\\
x^{2} +xz+ z^{2} &=& 16  \end{aligned} \right.$$ if, and only if, $$ \forall x,y,z \in \mathbb{R}^{+}: \left\{\begin{aligned} x^{2}+\sqrt{3} xy + y^2 &=& \color{blue}{5}^{2}\\
y^{2} + z^{2} &=& \color{blue}{3}^{2}\\
x^{2} +xz+ z^{2} &=& \color{blue}{4}^{2}  \end{aligned} \right. $$
Now, we can approach this problem as an algebraic geometry problem. Indeed, consider a triangle $\bigtriangleup XYZ$ with side lengths $3,4,5$ and draw a point $P$ inside the triangle such that $XP=x$, $YP=y$, and $ZP=z$. Now, you can considerer he equations in the context of the law of cosines.
Can you continue from here?
