A farmer prepares a rectangular vegetable garden of area 180 sq metres. With 39 metres of barbed wire, he can fence the three sides of the garden,

Let the length and breadth of the rectangular garden be $x$ and $y$ metre, respectively.

Given:

$x y=180$ sq m $\quad \ldots$ (i) and

$2 y+x=39$

$\Rightarrow x=39-2 y$

Putting the value of $x$ in (i), we get:

$(39-2 y) y=180$

$\Rightarrow 39 y-2 y^{2}=180$

$\Rightarrow 39 y-2 y^{2}-180=0$

$\Rightarrow 2 y^{2}-39 y+180=0$

$\Rightarrow 2 y^{2}-(24+15) y+180=0$

$\Rightarrow 2 y^{2}-24 y-15 y+180=0$

$\Rightarrow 2 y(y-12)-15(y-12)=0$

$\Rightarrow(y-12)(2 y-15)=0$

$\Rightarrow y=12$ or $y=\frac{15}{2}=7.5$

If $y=12, x=39-24=15$

If $y=7.5, x=39-15=24$

Thus, the length and breadth of the garden are $(15 \mathrm{~m}$ and $12 \mathrm{~m})$ or $(24 \mathrm{~m}$ and $7.5 \mathrm{~m})$, respectively.