Question

Out of a group of swans, $\frac{7}{2}$ times the square root of the number is playing on the shore of a tank. The two remaining ones are playing with the amorous fight, in the water. What is the total number of swans?

Answer 1

Let the total number of swans be $x$

Now, $x=\frac{7}{2}\sqrt{x}+2$

$\Rightarrow x-\frac{7}{2}\sqrt{x}-2=0$

Let $\sqrt{x}=y$

$\therefore y^2=x$

$\Rightarrow x-\frac{7}{2}\sqrt{x}-2=0$ becomes

$\Rightarrow y^2-\frac{7}{2}y-2=0$

$\Rightarrow 2y^2-7y-4=0$

$\Rightarrow 2y^2-8y+y-4=0$

$\Rightarrow 2y(y-4)+1(y-4)=0$

$\Rightarrow (2y+1)(y-4)=0$

$\Rightarrow 2y+1=0,y-4=0$

$\Rightarrow 2y=-1,y=4$

$\Rightarrow y=\frac{-1}{2},4$

$\Rightarrow \sqrt{x}=\frac{-1}{2},4$ where $y=\sqrt{x}$

$\Rightarrow \sqrt{x}=4$ and $x=\frac{1}{4}$ swans are meaningless

$\therefore x=16$ by squaring

Hence, the total number of swans is $16$