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,

Solution:

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.