Roots of an Equation - Changes of Sign
\[f(x)=0\]. 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.
\[x=2, : x^2-2x-1=2^2-2 \times 2-1=-1 \lt 0\]
\[x=3, : x^2-2x-1=3^2-2 \times 3-1=2 \gt 0\]
There is a sign change between
\[x=2, \: x=3\]so
\[2 \lt x \lt 3\].
We can narrow this interval.
\[x=2.5, \: x^2-2x-1=2.5^2-2 \times 2.5-1=0.25 \gt 0\]
Hence the sign change is for some
\[2.5 \lt x \lt 3\].