Question

If $x=\frac{1}{2-\sqrt{3}}$ then $x^3-3\left(x-1\right)\left(x+2\right)=$

• A. $6$
• B. $4$
• C. $5$
• D. $3$

Answer 1

Answer: (C)

$5$

We have,

$x=\frac{1}{2-\sqrt{3}}$

Then, $x^3-3(x-1)(x+2)=?$

Now,

$x=\frac{1}{2-\sqrt{3}}\times \frac{2+\sqrt{3}}{2+\sqrt{3}}onrationalize$

$x=\frac{2+\sqrt{3}}{4-3}$

$x=2+\sqrt{3}$

Then, put the equation,

$x^3-3(x-1)(x+2)$

$={(2+\sqrt{3})}^3-3(2+\sqrt{3}-1)(2+\sqrt{3}+2)$

$=2^3+{\left(\sqrt{3}\right)}^3+3\times 2\times \sqrt{3}(2+\sqrt{3})-3(1+\sqrt{3})(4+\sqrt{3})$

$=8+3\sqrt{3}+12\sqrt{3}+18-3(4+\sqrt{3}+4\sqrt{3}+3)$

$=26+15\sqrt{3}-3(7+5\sqrt{3})$

$=26+15\sqrt{3}-21-15\sqrt{3}$

$=5$