Question

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

Answer 1

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

To find $x^3+2x^2-7x+5$

$\therefore {\left(\frac{1}{2-\sqrt{3}}\right)}^3+2{\left(\frac{1}{2-\sqrt{3}}\right)}^2-7\left(\frac{1}{2-\sqrt{3}}\right)+5$

$=\frac{1}{26-15\sqrt{3}}+\frac{2}{7-4\sqrt{3}}-\frac{7}{2-\sqrt{3}}+5$

$=\frac{1336\sqrt{3}-2314}{1351-780\sqrt{3}}+5$

$=16\sqrt{3}+26+5$

$=16\sqrt{3}+31$