3. Given is the quadratic equation in $x$

$$
4(p-2)^{2} x^{2}-4(p-2) q x+q(4 p-3 q+4)-(p-1)^{2}=0, p, q \in \mathbb{N}
$$

What condition must $p$ and $q$ satisfy for the equation to have a double root? Find at least one double root that is a natural number.
Solution. For the given equation to be quadratic, it is necessary that $p-2 \neq 0$. It is directly verified that the discriminant of the equation is

$$
D=4(p-2)^{2}(2 q-p-1)^{2}
$$

For the equation to have a double root, it is necessary that $D=0$. Since $4(p-2)^{2} \neq 0$, it is necessary that $2 q-p-1=0$, which is the condition that $p$ and $q$ must satisfy for the quadratic equation to have a double root. That double root is

$$
x=\frac{2(p-2) q}{4(p-2)^{2}}=\frac{q}{2(p-2)}=\frac{p+1}{4(p-2)}
$$

For $x$ to be a natural number, it is necessary that $p \geq 3$ and $p+1 \geq 4(p-2)$, from which it follows that $p=3$ and $x=1$ is a double root.