Question & Answer
QUESTION

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

ANSWER Verified Verified
Hint: In this question if the total number of swans be \[x\] then the number of swans playing on the shore of the bank is $\dfrac{7}{2}\sqrt x $. Total number of swans will be equal to the summation of swans present in the different scenarios.

Complete step-by-step answer:

Let the total number of swan $ = x$
then the number of swan playing on the shore of a bank $ = \dfrac{7}{2}\sqrt x $
It's given remaining swan $ = 2$
Hence total number of swan = Number of swan playing on the shore of a bank plus number of remaining swan
Therefore equation can be written as:
\[\
   \Rightarrow x = \dfrac{7}{2}\sqrt x + 2 \\
   \Rightarrow x - 2 = \dfrac{7}{2}\sqrt x \\
   \Rightarrow 2\left( {x - 2} \right) = \dfrac{7}{2}\sqrt x \\
 \]
squaring both sides we get:
\[ \Rightarrow 4{\left( {x - 2} \right)^2} = 49x\]
applying the formula ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$ equation will be:
\[\
   \Rightarrow 4\left( {{x^2} + 4 - 4x} \right) = 49x \\
   \Rightarrow 4{x^2} + 16 - 16x = 49x \\
   \Rightarrow 4{x^2} - 65x + 16 = 0 \\
   \Rightarrow \left( {x - 16} \right)\left( {4x - 1} \right) = 0 \\
  \left( {x - 16} \right) = 0 \\
\]
or $\left( {4x - 1} \right) = 0$
$\therefore x = 16$ or $x = \dfrac{1}{4}$
Since the number of swans cannot be $\dfrac{1}{4}$
Therefore total number of swans $ = 16$

Note: In this question we calculated the total number of swans by adding the number of swans playing on the shore of a bank which is seven upon two times the square root of the number plus number of remaining swans in water. We formed the equation and after solving it we found the value of $x$ which we considered as the total number of swans i.e. sixteen.