, then by doing some figuring,
we might be able to squeeze down the size of the
interval (e.g. cut it in half) in which the root lies. That idea
is the heart of the bisection method.
The restriction under
which we operate is that 
and 
must have opposite signs --
one of them must be positive, the other negative. Then, because
f is continuous (we have to assume that), there must be a zero of
f somewhere in 
. Let c be the midpoint of
. Either c is the root, or the root lies in
or the root lies in 
.
Which happens? Well, we can evaluate
. If it is a zero (or at least if it is sufficiently small),
we have our root. If it is not zero, then
one pair of 
or 
have opposite signs.
Keep the half-interval with opposite signs. Repeat the process
until either 
f, evaluated at the midpoint of the interval,
is sufficiently small (less than a tolerance we must prescribe),
or 
the interval has been shrunk sufficiently small
(less than a second prescribed tolerance).
That is the bisection algorithm.