Question

between what two consecutive integers does square root of 379 lie?

Answer 1

say x is one number, next number (consecutive) is x+1
Product of these two will give 379\[x(x+1)=379 \rightarrow x^2 + x =379 \rightarrow x^2+x-379 =0\]
Solve fo x

Answer 2

https://www.wolframalpha.com/input/?i=complete%20the%20square%20x^2%2Bx-379&lk=2
roots are approx 19, 20. and, yes, they are the consecutive numbers.