Question

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

Answer 1

$x=\frac{1}{2-\sqrt{3}}=\frac{1}{2-\sqrt{3}}\times \frac{2+\sqrt{3}}{2+\sqrt{3}}=\frac{2+\sqrt{3}}{4-3}=2+\sqrt{3}$

$x-2=\sqrt{3}$

$(x-2{)}^2=3$

$x^2-4x+4=3$

$x^2-4x+1=0$

Therefore,

$x^3-2x^2-7x+5=x(x^2-4x+1)+2(x^2-4x+1)+3$

$=x\times 0+2\times 0+3=3$