Question

If $x=\frac{1}{2-\sqrt{3}}$, then the value of $x^3-2x^2-7x+5$ is

• A. $2$
• B. $1$
• C. $0$
• D. $3$

Answer 1

Answer: (D)

$3$

Rationalize and compute the value of given expression:

$x=\frac{1}{2-\sqrt{3}}=\frac{(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})}=\frac{2+\sqrt{3}}{1}=2+\sqrt{3}{x}^2={(2+\sqrt{3})}^2=7+4\sqrt{3}{x}^3=x^2.x=(2+\sqrt{3}).(7+4\sqrt{3})=(14+8\sqrt{3}+7\sqrt{3}+12)=26+15\sqrt{3}{x}^3-2x^2-7x+5=26+15\sqrt{3}-2(7+4\sqrt{3})-7(2+\sqrt{3})+526+15\sqrt{3}-14-8\sqrt{3}-14-7\sqrt{3}+5=17-14=3.$