Out of a group of swans, 72 times the square roots of the total number are playing on the shore of a pond. The remaining two are swimming in water. Find the total number of swans.

# Out of a group of swans, $\frac{7}{2}$ times the square roots of the total number are playing on the shore of a pond. The remaining two are swimming in water. Find the total number of swans.

1. A
0
2. B
4
3. C
16
4. D
8

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### Solution:

Concept- Using the method of quadratic factorization by splitting the middle term then we will apply the conditions given in the equation.
Given $\frac{7}{2}$ times of the square root of the total number of swans $\sqrt{n}$, i.e.
Adding the remaining two swans that are swimming, then we get,
$⇒\frac{7}{2}\sqrt{n}+2=n$
$⇒\frac{7}{2}\sqrt{n}=n-2$
On squaring both sides, we get

By using identity , we get
$⇒49n=4{n}^{2}-16n+16$
$⇒4{n}^{2}-65n+16=0$
By using the method of factorization, and splitting the middle term, we will get,
$⇒4{n}^{2}-n-64n+16=0$

Taking ($4n-1$) as common, we get

Now, either the values of $n=\frac{1}{4}$ or $n=16$
As, the number of swans cannot be in the form of fraction $=\frac{1}{4}$, there the correct answers $=16$
The total number of swans = 16
Hence, the correct answer is option 3.

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