What maths topic is this?

I'd call it elementary algebra.

I don't agree with "RealPro" who says estimates are done with Newton's method. On the contrary, Newton's method is used AFTER you have an estimate. Newton's method is calculus.

The estimate can come from graphing f(x), which is high-school algebra, unless the function f(x) itself is beyond the scope of high-school algebra. For example, one might not consider the trigonometric functions to be a part of high-school algebra...you'd call them a part of "trigonometry."

It's called "algebra".

Maybe calculus.

Estimates are most often done using Newton's method, or eyeing it in case you were given the graph.

If a function can be easily approximated with another function then you can also do that, keeping in mind the limitations i.e. the interval where it satisfies the approximation.

For example when f(x) = sqrt(1+x^2), then for large x (greater than 10 or whatever, depending on the error allowed) you can approximate f(x) = x.

So if f(x) = sqrt(1+x^2) - 69, then x=69 would be a reasonable estimate for the root.