Let's determine $y$ such that the roots of the equation

$$
x^{2}-3 x y+y^{2}+2 x-9 y+1=0
$$

are equal ($x_{1}$ and $x_{2}$); for which values of $y$ are the roots of the equation real? Solve the equation for $y$ and determine $x$ such that the roots of the equation are real.
$1^{\circ}$. The roots of the given equation are equal if the discriminant is 0, that is, if:

$$
5 y^{2}+24 y=y(5 y+24)=0
$$

from which

$$
y_{1}=0, y_{2}=-\frac{24}{5}
$$

$2^{\circ}$. The roots are real if the discriminant is positive; that is, if:

$$
y(5 y+24) \geqq 0
$$

from which

$$
y_{1} \geqq 0, y_{2} \leqq-\frac{24}{5}
$$

$3^{\circ}$. The given equation can also be written as:

$$
y^{2}-(3 x+9) y+x^{2}+2 x+1=0
$$

from which

$$
y=\frac{3 x+9}{2} \pm \frac{1}{2} \sqrt{5 x^{2}+46 x+77}
$$

$4^{\circ}$. The roots of the quadratic equation in $y$ are real if

$$
5 x^{2}+46 x+77=(x+7)(5 x+11) \geqq 0
$$

this product is $\geqq 0$ if

$$
x \geqq-7 \text { and } x \geqq-\frac{11}{5}, \text { that is, if } x \geqq-\frac{11}{5}
$$

or

$$
x \leqq-7 \text { and } x \leqq-\frac{11}{5}, \text { that is, if } x \leqq-7
$$

(Solved by: Fekete Mihály, Zenta.)

The problem was also solved by: Bánó L., Ehrenfeld N., Erdélyi I., Esztó P., Fodor H., Földes R., Freund E., Füstös P., Heimlich P., Jánosy Gy., Kiss E., Paunz A., Pichler S., Schuster Gy., Szilas O.

The problem was partially solved by: Arató I., Bauer E., Bayer N., Blum J., Chambré M., Cukor G., Epstein K., Erdős V., Gádor K., Kirchknopf E., Kovács Gy., Krémusz R., Neumann L., Pető L., Rosenthal M., Sárközy P., Spitzer L., Tandlich E., Tóth B., Vilcsek A., Mathematikai kör, Bp. V. ker. áll. főgymn.