Question

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

• A. $3$
• B. $2$
• C. $5$
• D. $4$

Answer 1

The correct option is D $4$

$x=\frac{1}{\sqrt{5}-2}$
$x=\frac{1}{\sqrt{5}-2}\times \frac{\sqrt{5}+2}{\sqrt{5}+2}=\frac{\sqrt{5}+2}{5-4}=\sqrt{5}+2$

$\therefore x-2=\sqrt{5}$
Squaring both sides, we have

$x^2-4x+4=5$

$x^2-4x-1=0$....(i)
Now, $x^3-3x^2-5x+3=(x^2-4x-1)(x+1)+4$
$=0(x+1)+4=4$ [Using (i)]