Let the roots of the quadratic equation $x^{2}-p^{2} x+p q=0$ be 1 greater than the roots of the quadratic equation $x^{2}+p x+q=0$. Determine the roots, as well as the values of $p$ and $q$.
I. solution: Denoting the roots of the second equation by $x_{1}, x_{2}$, the roots of the first equation are $x_{1}+1$ and $x_{2}+1$. Thus, based on the relationships between the roots and the coefficients, we can write four equations for the unknowns $x_{1}, x_{2}, p$, and $q$:

$$
x_{1}+x_{2}=-p
$$

$$
\begin{gathered}
x_{1} x_{2}=q \\
x_{1}+x_{2}+2=p^{2} \\
\left(x_{1}+1\right)\left(x_{2}+1\right)=x_{1} x_{2}+\left(x_{1}+x_{2}\right)+1=p q
\end{gathered}
$$

Considering (1) and (2), from (3) and (4) we get the following system of equations for $p$ and $q$:

$$
-p+2=p^{2}
$$

From (5), $p_{1}=-2, p_{2}=1$, and from (6), after rearrangement, we get

$$
(p-1)(q+1)=0
$$

For $p_{1}$, $q_{1}=-1$, and for $p_{2}$, the equation holds for any $q$.

In the first case, the given equations are: $x^{2}-4 x+2=0$ and $x^{2}-2 x-1=0$, with roots $2 \pm \sqrt{2}$ and $1 \pm \sqrt{2}$, respectively, and these indeed satisfy the requirement.

The same holds for the second case with the equations $x^{2}-x+q=0$ and $x^{2}+x+q=0$, with roots $(1 \pm \sqrt{1-4 q}) / 2$ and $(-1 \pm \sqrt{1-4 q}) / 2$, respectively. These are real if and only if $1-4 q \geqq 0$, i.e., $q \leqq 1 / 4$.

Dániel Gábor (Bp. VIII., Piarista g. IV. o. t.)

II. solution: We can also arrive at the system of equations (5), (6) through a graphical consideration. The left sides of the given equations represent two parabolas that are congruent because the absolute value of the coefficient of $x^{2}$ is 1 in both, and they are of the same orientation because the signs of these coefficients are the same, with the vertex of both parabolas pointing downwards.

![](https://cdn.mathpix.com/cropped/2024_05_02_8a0484844274195f38cdg-1.jpg?height=545&width=634&top_left_y=1522&top_left_x=735)

Thus, they can be obtained from each other by a (parallel) translation. The magnitude of the translation is 1 unit along the $X$-axis due to the requirement on the roots, which are the $X$-intercepts of the parabolas and the $X$-axis. Specifically, the parabola representing the left side of the first equation is translated 1 unit to the right relative to the other (see the first figure). Then, the function on the left side of the first equation can also be written as $y=(x-1)^{2}+p(x-1)+q=x^{2}-(2-p) x+(1-p+q)$, and this is identical to the function $y=x^{2}-p^{2} x+p q$ if and only if the coefficients of the terms are equal. The system (5), (6) precisely demands this, the equality of the coefficients of the linear term and the constant term.