Grandma had a square garden. She bought a few neighboring plots, thus obtaining a square plot whose side was three meters longer than the side of the original garden. The area of the new plot was nine square meters larger than twice the original area.

How long was the side of the original garden?

(K. Buzáková)
Let $a$ be the length of the side of the original square garden. After the purchase, a new square plot was formed, with the side length of $a+3$. According to the areas of the plots given in the problem, we have

$$
(a+3)^{2}=2 a^{2}+9 \text {. }
$$

By equivalent transformations, we get

$$
\begin{aligned}
a^{2}+6 a+9 & =2 a^{2}+9 \\
6 a & =a^{2} \\
0 & =a(a-6)
\end{aligned}
$$

The given equation has two solutions: $a=0$ and $a=6$.

The original garden had non-zero dimensions, so its side was six meters long.

Note. Although we do not know what the purchased plots looked like, we can illustrate the areas of the gardens (and their auxiliary divisions) as follows:

![](https://cdn.mathpix.com/cropped/2024_04_17_e6d56f2031023e9db813g-1.jpg?height=482&width=488&top_left_y=1575&top_left_x=790)

The large square is composed of two squares and two congruent rectangles, or one square and four congruent rectangles. This visual representation reveals the result $a=3+3=6$.

Evaluation. 2 points for formulating the conditions from the problem using one variable; 2 points for equivalent transformations; 2 points for excluding the zero solution and concluding.

Solutions based on graphical representation should be evaluated according to the quality of the accompanying commentary.