Roots of an Equation - Changes of Sign

Suppose we are to solve an equation  
\[f(x)=0\]
. We may not be able to solve the equation exactly, but we may be able to say that the solution lies between two numbers  
\[x_1, \: x_2\]
  if  
\[f(x)\]
  changes sign between these values of  
\[x\]
. We may be able to narrow the interval by finding smaller intervals on which there is a sign change.
Suppose  
\[x^2-2x-1=0\]
,
When  
\[x=2, : x^2-2x-1=2^2-2 \times 2-1=-1 \lt 0\]

When  
\[x=3, : x^2-2x-1=3^2-2 \times 3-1=2 \gt 0\]

There is a sign change between  
\[x=2, \: x=3\]
  so  
\[x^2-2x-1=0\]
  for some  
\[2 \lt x \lt 3\]
.
We can narrow this interval.
When  
\[x=2.5, \: x^2-2x-1=2.5^2-2 \times 2.5-1=0.25 \gt 0\]

Hence the sign change is for some  
\[2.5 \lt x \lt 3\]
.