A1. If $r$ is a number for which $r^{2}-6 r+5=0$, what is the value of $(r-3)^{2}$ ?
Solution: The answer is 4 .

Solution 1: Note that $(r-3)^{2}=r^{2}-6 r+9$. Since $r^{2}-6 r+5=0, r^{2}-6 r+9=4$. Therefore, the answer is 4 .

Solution 2: The quadratic equation $r^{2}-6 r+5$ factors as
$$
(r-1)(r-5) .
$$

Therefore, $r=1$ or $r=5$. If $r=1$, then $(r-3)^{2}=(-2)^{2}=4$. If $r=5$, then $(r-3)^{2}=2^{2}=4$. In either case, $(r-3)^{2}=4$.

Solution 3: By completing the square on $r^{2}-6 r+5$, we have $r^{2}-6 r+5=(r-3)^{2}-4$. Since $r^{2}-6 r+5=0,(r-3)^{2}-4=0$. Hence, $(r-3)^{2}=4$.