Question

If $\frac{5-\sqrt{3}}{2+\sqrt{3}}=x+y\sqrt{3},$ then

• A. x = 13, y = -7
• B. x = -13, y = 7
• C. x = -13, y = -7
• D. x = 13, y = 7

Answer 1

The correct option is A

x = 13, y = -7

$\frac{5-\sqrt{3}}{2+\sqrt{3}}=x+y\sqrt{3}$

$\frac{5-\sqrt{3}}{2+\sqrt{3}}=\frac{(5-\sqrt{3})(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}$

(Rationalising the denominator)

$=\frac{10-5\sqrt{3}-2\sqrt{3}+3}{(2{)}^2-(\sqrt{3}{)}^2}=\frac{13-7\sqrt{3}}{4-3}$

$=\frac{13-7\sqrt{3}}{1}=13-7\sqrt{3}$

$\therefore 13-7\sqrt{3}=x+y\sqrt{3}$

Comparing, we get

x = 13, y = -7