Question

The difference between squares of two numbers is 120. The square of smaller number is twice the greater number. Find the numbers.

Answer 1

Let the smaller number be x.
Larger number be y.
The square of smaller number is twice the greater number.
$x^2=2y$
The difference between squares of two numbers is 120.
$$\begin{gathered} y^2-x^2=120 \\ \Rightarrow y^2-2y-120=0 \\ \Rightarrow y^2-12y+10y-120=0 \\ \Rightarrow y\left(y-12\right)+10\left(y-12\right)=0 \\ \Rightarrow \left(y+10\right)\left(y-12\right)=0 \\ \Rightarrow y=12,-10 \end{gathered}$$
So, the numbers are 12 as the larger number and the smaller number is
$$\begin{gathered} x^2=2\times 12=24 \\ \Rightarrow x=\pm \sqrt{24} \end{gathered}$$