Question

If $x=-5+2\sqrt{-4}$, then find the value of $x^4+9x^3+35x^2-x+4$.

Answer 1

Since $x=-5+2\sqrt{-4}$

$\Rightarrow x=-5+(2i\times 2)$

$\Rightarrow x=-5+4i$

$\Rightarrow x+5=4i$

$\Rightarrow {(x+5)}^2={\left(4i\right)}^2$ (Squaring both sides)

$\Rightarrow x^2+10x+25=16i^2$

$\Rightarrow x^2+10x+25=-16$

$\Rightarrow x^2+10x+25+16=0$

$\Rightarrow x^2+10x+41=0$.

$\therefore x^4+9x^3+35x^2-x+4=(x^2+10x+41)(x^2-x+4)-160$

$=0-160=-160$.

Hence, the required value is $160$.