Let's determine the coordinates of the center of the circle given by the equation

$$
x^{2}+y^{2}-2 p x-2 q y+2 q-1=0
$$

such that the abscissas of the intersection points of the circle with the $X$-axis are equal to the roots of the equation

$$
x^{2}+b x+c=0
$$

How can we use the found result to construct the roots of the quadratic equation?
I. Equation (1) can be written as:

$$
(p-x)^{2}+(q-y)^{2}=p^{2}+q^{2}-2 q+1=p^{2}+(q-1)^{2}
$$

According to this, the center of the circle is the point $K(p, q)$, and the square of its radius is $r^{2}=p^{2}+(q-1)^{2}$, which is the square of the distance between $K$ and the point $C(0 ; 1)$. Therefore, the circle passes through the fixed point $C$ for any values of $p$ and $q$.

The coordinates of the intersection points of the circle and the $X$-axis can be obtained by substituting $y=0$ into equation (1):

$$
x^{2}+2 p x+2 q-1=0
$$

The coefficient of the quadratic term in this equation matches that in (2), so their roots will coincide if and only if the other coefficients also match:

$$
-2 p=b \quad \text { and } \quad 2 q-1=c
$$

From this, the coordinates of $K$ are:

$$
p=-b / 2, \quad q=(1+c) / 2
$$

II. According to this, the roots of (2) can be obtained by drawing a circle through point $C$ around the point $K(-b / 2,(1+c) / 2)$ and reading the abscissas of the points of intersection with the $X$-axis. There will certainly be a common point if $K$ is below the line $x=1 / 2$, or exactly on the line, i.e., when $q \leqq 1 / 2, c \leqq 0$. If $q>1 / 2$, the condition for the existence of a common point is that the $KC$ radius is at least as large as the distance of $K$ from the $X$-axis, or the ordinate of $K$. From this,

$$
\sqrt{\left(-\frac{b}{2}\right)^{2}+\left(\frac{1+c}{2}-1\right)^{2}} \geqq \frac{1+c}{2}
$$

In other words, since both sides are positive, $b^{2}-4 c \geqq 0$.

This condition also includes the above requirement $c \leqq 0$. The condition expresses that $K$ is an external point or lies on the parabola defined by $C$ as the focus and the $X$-axis as the directrix.

Agnes Dévai (Sárvár, Tinódi S. g. III. o. t.)

Note. In practice, this method is often applied using a two-point (measuring) compass, with the moving pin of the compass inserted into the intersection point with the $X$-axis during the reading.