4. If $a$ and $b$ are integers and $\sqrt{7-4 \sqrt{3}}$ is one of the roots of the equation $x^{2}+a x+b=0$, find the value of $a+b$.
(A) -3
(B) -2
(C) 0
(D) 2
(E) 3
4. Answer: (A)
Note that $\sqrt{7-4 \sqrt{3}}=\sqrt{(2-\sqrt{3})^{2}}=2-\sqrt{3}$. Since $\sqrt{7-4 \sqrt{3}}$ is a root of the equation $x^{2}+a x+b=0$, we have $(7-4 \sqrt{3})+(2-\sqrt{3}) a+b=0$. Rearranging the terms, we obtain $(7+2 a+b)-(4+a) \sqrt{3}=0$. This implies that $7+2 a+b=0$ and $4+a=0$, which give $a=-4$ and $b=1$. Thus $a+b=-3$.