10. It is given that $a, b$ and $c$ are three real numbers such that $a+b=c-1$ and $a b=c^{2}-7 c+14$. Find the largest possible value of $a^{2}+b^{2}$.
(A) 5
(B) 6
(C) 8
(D) 9
(E) 10
10. Answer: (C)
The roots of the equation $x^{2}-(c-1) x+c^{2}-7 c+14=0$ are $a$ and $b$, which are real. Thus the discriminant of the equation is non-negative. In other words,
$$
(c-1)^{2}-4\left(c^{2}-7 c+14\right)=-3 c^{2}+26 c-55=(-3 c+11)(c-5) \geq 0 .
$$

So we have $\frac{11}{3} \leq c \leq 5$. Together with the equalities
$$
\begin{aligned}
a^{2}+b^{2} & =(a+b)^{2}-2 a b \\
& =(c-1)^{2}-2\left(c^{2}-7 c+14\right) \\
& =-c^{2}+12 c-27=9-(c-6)^{2}
\end{aligned}
$$
we see that maximum value of $a^{2}+b^{2}$ is 8 when $c=5$.