Let the two numbers are x and y.

According to the question,

Sum of squares fo two numbers = 97

ie., \( \Large x^{2} + y^{2} = 97 \) ... (i)

and the square of their difference = 25

ie., \( \Large  \left(x - y\right)^{2} = 25   \) .. (ii)

=> \( \Large x - y = 5 \) .... (iii)

From eqn (ii)

\( \Large  \left(x^{2} + y^{2}\right)-2xy = 25  \)

=> 97 - 2xy = 25 (From eqn 1)

=> 2xy = 72

=< xy = 36 ... (iv)

Now, we have

\( \Large  \left(x + y\right)^{2}=  \left(x^{2} + y^{2} \right) + 2xy   \)

= 97 + 72 = 169

\( \Large x + y  =13 \) ... (V)

Now, from eqns (III) and (V) , we get

2x = 18

x = 9 and y = 4

The product of both the numbers  = xy = 9 X 4 = 36