Step 1: Picture the triangle.
The two roots $z_1,z_2$ and the origin $O$ form an equilateral triangle. So all three sides are equal, and the angle at $O$ between $z_1$ and $z_2$ is $60^\circ$.
Step 2: Equal lengths.
Sides $Oz_1$ and $Oz_2$ are equal, so $|z_1|=|z_2|=r$ for some $r$. The two roots have the same distance from the origin.
Step 3: Use the root rules.
For $z^2+az+b=0$, the sum of roots is $z_1+z_2=-a$ and the product is $z_1z_2=b$.
Step 4: Product gives $b$.
Taking sizes, $|z_1z_2|=|z_1||z_2|=r\cdot r=r^2$. So $|b|=r^2$, and we write $b=r^2$ (size form).
Step 5: Sum gives $a$.
Two equal vectors with $60^\circ$ between them add to a vector of length $2r\cos\big(\tfrac{60^\circ}{2}\big)=2r\cos30^\circ=\sqrt3\,r.$ So $|a|=\sqrt3\,r$, hence $a^2=3r^2.$
Step 6: Combine.
Since $r^2=b$, we get $a^2=3r^2=3b$. \[ \boxed{a^2=3b} \]