Call the smaller number x.
Then the bigger number is x + 1

Sum of squares is 1105

x² + (x + 1)² = 1105

x² + x² + 2x + 1 = 1105

2x² + 2x + 1 = 1105

2x² + 2x - 1104 = 0

x² + x - 552 = 0

Doing your cross factorisation, you get

(x + 24)(x - 23) = 0

x = -24 (rejected as x is positive) or x = 23

So the smaller number is 23
The bigger one is 23 + 1 = 24

Trick to cross factorise for such large numbers :


Use prime factorisation (learnt in sec 1) to break the number down first.

552 = 2³ × 3 × 23

You can then pair 2³ × 3 = 8 × 3 = 24

Thank you