suppose we change the right-hand side of the equation (a)^2=(b)^2 + 25 with (b-1)^2 + 25 then, would the Left-hand side still be equal to the Right-hand side?

If not, then what is the reason?

Also, what changes do we need to make to the Left-hand side in order to make Left-hand side = Right-hand side?

The equation is only true for certain pairs of values (a,b), not for all pairs. If you replace b with (b-1), then it will be true for a DIFFERENT set of pairs. So the answer to your question, given a specific pair (a,b), is NO: for that pair, the equation will no longer be true.

The reason is that you have changed the role b plays in the equation, which can be interpreted as changing the definition of b. If the original equation is true when b has a specific value, the new equation will be true when b-1 has that value -- that is, when the "new b" is 1 more than the "old b".

There is nothing you can do to the left side so that the equation will still be true for the same set of pairs (that is, will have the same solution set) as the original, unless it involves b as well as a.

But you can gain an understanding of this by "playing" with it. Find some pairs (a,b) that work, such as (13,12) and (13,-12), or various non-integer pairs; then compare these to pairs for which the new equation works, such as (13,13) and (13,-11). See for yourself that (13,12) does not satisfy the new equation.