## 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
$2.5 \lt x \lt 3$
.