Find the two natural numbers which differ by 5 and the sum of whose squares is 97.
Let the two numbers be x and x + 5.
From the given information,
x2+(x+5)2=97
2×x2+10×x+25–97=0
2×x2+10×x–72=0
x2+5×x–36=0
(x + 9) (x – 4) = 0
x = -9 or 4
Since, -9 is not a natural number. So, x = 4.
Thus, the numbers are 4 and 9.