Question

Does r^2 - 10r = -22 have a zero between 6 and 7?

Answer 1

Solution

$f(r)=r^2-10r+22$

Using

Intermediate Value Theorem,

When we have two points connected by a continuous curve:

one point below the line the other point above the line then there will be at least one place where the curve crosses the line!

$\begin{gathered} \text{ Since, } \\ f(6)=6^2-10(6)+22=-2 \\ \\ f(7)=7^2-10(7)+22=1 \\ \\ \text{ since }f(6)\cdot f(7)<0 \\ \\ \Rightarrow\text{ }\exists\text{ }r\in(6,7)\text{ such that }f(r)=0 \end{gathered}$