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One-fourth of a herd of camels was seen in the forest. Twice the square root of the herd had gone to the mountains and the remaining 15 camels were seen on the bank of a river. Find the total number of camels.

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Hint: Let us denote a variable $x$ which represents the total number of camels that are present in the herd. Read the question line by line and apply all the conditions that are given in the question on this variable $x$ to get an equation. The equation can be solved and we can find the value of $x$.

In this question, of the total number of camels in a herd, one-fourth of them are in the forest, twice the square root of the total camels had gone to the mountains and the remaining 15 were seen on the bank of the river. We are asked to find the total number of camels that are present in the herd.
Let us denote a variable $x$ which represents the total number of camels that are present in the herd.
It is given that one-fourth of these camels are in the forest. So, the number of camels in the forest is equal to $\dfrac{1}{4}x..........\left( 1 \right)$.
It is given that twice the square root of the herd had gone to the mountains. So the number of camels that had gone to the mountains is equal to $2\sqrt{x}........\left( 2 \right)$.
Also, it is given that the number of camels that were seen on the bank of the river is equal to $15.......\left( 3 \right)$.
The sum of the terms which we got in equation $\left( 1 \right),\left( 2 \right),\left( 3 \right)$ should be equal to the total number of the camels i.e. $x$. This means,
$\begin{align}
  & \dfrac{1}{4}x+2\sqrt{x}+15=x \\
 & \Rightarrow 2\sqrt{x}=x-\dfrac{1}{4}x-15 \\
 & \Rightarrow 2\sqrt{x}=\dfrac{3}{4}x-15 \\
 & \Rightarrow 2\sqrt{x}=\dfrac{3x-60}{4} \\
 & \Rightarrow 8\sqrt{x}=3x-60 \\
\end{align}$
Squaring both the sides, we get,
$\begin{align}
  & 64x=9{{x}^{2}}+3600-360x \\
 & \Rightarrow 9{{x}^{2}}-424x+3600=0 \\
\end{align}$
To solve this quadratic equation, we will use the quadratic formula. Let us assume a quadratic equation $a{{x}^{2}}+bx+c=0$. From the quadratic formula, the roots of this equation are given by,
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Using quadratic formula in the equation $9{{x}^{2}}-424x+3600=0$, we get,
\[\begin{align}
  & x=\dfrac{-\left( -424 \right)\pm \sqrt{{{\left( -424 \right)}^{2}}-4\left( 9 \right)\left( 3600 \right)}}{2\left( 9 \right)} \\
 & \Rightarrow x=\dfrac{-\left( -424 \right)\pm \sqrt{{{\left( -424 \right)}^{2}}-4\left( 9 \right)\left( 3600 \right)}}{2\left( 9 \right)} \\
 & \Rightarrow x=\dfrac{424\pm \sqrt{179776-129600}}{18} \\
 & \Rightarrow x=\dfrac{424\pm \sqrt{50176}}{18} \\
 & \Rightarrow x=\dfrac{424\pm 224}{18} \\
 & \Rightarrow x=36,\dfrac{100}{9} \\
\end{align}\]
Since $x$ represents the number of camels, it must be an integer. So, $x=36$.
Hence, the total number of camels is equal to $36$.

Note: There is an alternative way to solve the quadratic equation $8\sqrt{x}=3x-60$. We can assume a variable $p$ and then we can substitute $x={{p}^{2}}$ in this quadratic equation and then solve for $p$ using a quadratic formula. Later, we can re-substitute ${{p}^{2}}=x$ or $p=\sqrt{x}$ and then solve to get $x$.
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