Solve quadratic related problem without brute force/trial and error or quadratic formula.

Question

At least I need to explain to an 11 year old

One square is cut out of another. Side lengths of each square are whole numbers less than 25. Remaining area of larger is 57. What is perimeter of smaller square?

Obviously we can use excel to find B (bigger), L (littler, if that's even a word) s.t $B^2 - L^2 = 57$ namely B=11,L=8 but I think there is some simple trick to this that I'm overlooking like we would be able to compute 4L without directly computing B or L. Completing the square?

Like

$B^2 - 57 - 4L + 4 = L^2 - 4L + 4$

Side question: is quadratic formula even helpful here? I think we are still going to do trial and error

Answer 1

Hint : Use $$B^2-L^2=(B-L)(B+L)=57$$ For every pair $(u,v)$ with $uv=57$, solve the linear equation system $$B-L=u$$ $$B+L=v$$ This should be faster than trial and error.

Answer 2

From what I understand, we have:

$B^{2} -L^{2} = 57 \Rightarrow (B+L)(B-L) = 57$

Factorising $57$: $57 = 19 \cdot 3$

Since $B,L \in \mathbb{N}$, and $B>L$, we get $B+L = 19, B-L = 3$

This gives us $B = 11$ and $L = 8$. So the perimeter of the smaller square is $32$

Edit: note we also get the solution $B = 29, L = 28$, but we reject these using the initial condition $B,L < 25$