Question

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

Answer 1

Hint: To solve these types of questions, we should start by taking the total number of swans as $x$. We have to apply the operations mentioned in the question, step by step. The first step in taking the square root, which will give $\sqrt{x}$. The second step is multiplying $\dfrac{7}{2}$ to the $\sqrt{x}$, which we can get as $\dfrac{7}{2}\sqrt{x}$. We can infer from the question that from the original number of swans $x$, $\dfrac{7}{2}\sqrt{x}$ swans are at the shore and the remaining 2 are inside water. So the mathematical expression will be, $x=\dfrac{7}{2}\sqrt{x}+2\Rightarrow x-2=\dfrac{7}{2}\sqrt{x}$. Squaring on both sides and solving the quadratic equation, gives the answer of $x$.

Complete step-by-step answer:
Let us assume the number of swans as $x$. We have to understand the operations given in the question and we should apply them to the value of $x$.
Let us consider the statement $\dfrac{7}{2}$ times the square root of the total number of swans are playing on the shore of a tank.
First, we should apply the square root and then multiply the term by $\dfrac{7}{2}$.
Step-1, applying the square root, we get $\sqrt{x}$
Step-2, multiplying by $\dfrac{7}{2}$, we get $\dfrac{7}{2}\sqrt{x}$.
So, the number of swans at the shore are $\dfrac{7}{2}\sqrt{x}$.
We can infer from the question that the remaining swans are 2 in number and they are deep inside water.
We can infer that the total number of swans is the sum of the swans at the shore and the swans in deep water.
We can write it as
$\begin{align}
  & x=\dfrac{7}{2}\sqrt{x}+2 \\
 & x-2=\dfrac{7}{2}\sqrt{x} \\
 & 2x-4=7\sqrt{x} \\
\end{align}$
Squaring on both sides, we get
\[\begin{align}
  & {{\left( 2x-4 \right)}^{2}}={{\left( 7\sqrt{x} \right)}^{2}} \\
 & 4{{x}^{2}}-16x+16=49x \\
 & 4{{x}^{2}}-65x+16=0 \\
\end{align}\]
We can factorise the above expression as
$\begin{align}
  & 4{{x}^{2}}-64x-x+16=0 \\
 & 4x\left( x-16 \right)-1\left( x-16 \right)=0 \\
 & \left( 4x-1 \right)\left( x-16 \right)=0 \\
\end{align}$
We get the values of x as
$x=\dfrac{1}{4}\text{ or }16$
As the number of swans cannot be a fraction, we can conclude that the total number of swans is 16.
$\therefore $ The total number of swans is 16

Note: Students should be careful while squaring the terms. The main trick while squaring the terms is that the term which has a square root term should be on one side and the remaining terms on the other side. In the question, if we square on both sides the equation $x=\dfrac{7}{2}\sqrt{x}+2$, we get an equation as
$\begin{align}
  & {{x}^{2}}={{\left( \dfrac{7}{2}\sqrt{x}+2 \right)}^{2}} \\
 & {{x}^{2}}=\dfrac{49}{4}x+4+14\sqrt{x} \\
\end{align}$
which is not solvable directly. So, it is always advisable to square on both sides keeping the square root term on one side and the other terms on the other side.
.