Let's calculate the difference between the roots of the following equation.

$$
x^{2}-2\left(a^{2}+b^{2}+c^{2}-2 a c\right) x+\left(b^{2}-a^{2}-c^{2}+2 a c\right)^{2}=0
$$
If the roots of the equation $x^{2}+p x+q=0$ are $x_{1}$ and $x_{2}$, then

$$
\left(x_{1}-x_{2}\right)^{2}=x_{1}^{2}-2 x_{1} x_{2}+x_{2}^{2}=\left(x_{1}+x_{2}\right)^{2}-4 x_{1} x_{2}=p^{2}-4 q
$$

Therefore,

$$
x_{1}-x_{2}= \pm \sqrt{p^{2}-4 q}
$$

Substituting the given values, we get

$$
x_{1}-x_{2}= \pm 4 b(a-c)
$$

(Vilmos Erdös, Budapest.)

Number of solutions: 71.