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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.

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answer is C.

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Detailed 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,

\Rightarrow \frac{7}{2} \sqrt{n} + 2 = n

\Rightarrow \frac{7}{2} \sqrt{n} = n - 2

On squaring both sides, we get

\Rightarrow 7^{2} n = 4 (n - 2)^{2}

\Rightarrow 49 n = 4 (n^{2} - 4 n - 4)

By using identity

(a - b)^{2} = a^{2} + b^{2} - 2 ab

, we get

\Rightarrow 49 n = 4 n^{2} - 16 n + 16

\Rightarrow 4 n^{2} - 65 n + 16 = 0

By using the method of factorization, and splitting the middle term, we will get,

\Rightarrow 4 n^{2} - n - 64 n + 16 = 0

\Rightarrow n (4 n - 1) - 16 (4 n - 1) = 0

Taking (

4 n - 1

) as common, we get

\Rightarrow (4 n - 1) (n - 16) = 0

Now, either the values of

n = \frac{1}{4}

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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