Question

The sum of a natural number and its positive square root is $132$. Find the number.

Answer 1

suppose requires natural number $=x$

sum of natural number and its square root $=x+\sqrt{x}$

putting

$\begin{array}{c}\sqrt{x}=y\\ x=y^2\end{array}$

$\begin{array}{c}{y}^2+y=132\\ y^2+12y-11y-132=0\\ y\left(y+12\right)-11\left(y+12\right)=0\\ \left(y+12\right)\left(y-11\right)\\ y=11\\ \sqrt{x}=11\\ x={\left(11\right)}^2\\ x=121\end{array}$