Question

Out of a group of Swans, $\dfrac{7}{2}$ times the square root 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

Answer 1

Hint:In order to solve this question, we will use the method of quadratic factorization by splitting the middle term then we will apply the conditions given in the question.

Complete step-by-step answer:
Let the total number of swans be ‘$n$ ’
It is given that $\dfrac{7}{2}$ times of the square root of the total number of swans that is,
$\dfrac{7}{2}\sqrt n $
Now, if we add the remaining two swans that are swimming, then the equation will be,
$\dfrac{7}{2}\sqrt n + 2 = n$
We can also write it as
$\dfrac{7}{2} \sqrt n = n - 2$
Squaring, both sides we will get
${7^2}n = 4{\left( {n - 2} \right)^2}$
$49n = 4\left( {{n^2} - 4n + 4} \right)$ (By using the identity ${\left( {a - b} \right)^2}$ which is equal to ${a^2} + {b^2} - 2ab$)
$49n = 4{n^2} - 16n + 16$
$ \Rightarrow $ $4{n^2} - 65n + 16 = 0$ ($65n = $ $ - 16n + 49n$ )
By using the method of factorization, and splitting the middle term we will get,
$
  {\text{ }}4{n^2} - n - 64n + 16 = 0 \\
   \Rightarrow n\left( {4n - 1} \right) - 16\left( {4n - 1} \right) = 0 \\
 $
Taking $\left( {4n - 1} \right)$ as common, we will get
$\left( {4n - 1} \right)\left( {n - 16} \right) = 0$
Now the values of $n$,
(i), value of $n$
 $
  4n - 1 \\
  n = \dfrac{1}{4} \\
 $
(ii) value of $n$
$
  n = 16 \\
 $
We know that number of swans cannot be in the form of fraction that is $\dfrac{1}{4}$, Therefore, the right answer is 16
That is the total number of swans are 16.

Note- Whenever we come up with this type of question, one must know the basic identities to solve this kind of question like ${\left( {a + b} \right)^2},{\left( {a + b} \right)^3},\left( {a + b} \right)\left( {a - b} \right)$ etc. By using these simple identities and methods of quadratic equations one can easily solve this question.