Question

Find the value of $x^3+9x^2-x+4$ for $x=-5+2\sqrt{-4}$

Answer 1

Consider the given expression.

$\Rightarrow x^3+9x^2-x+4$

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

Therefore,

$\Rightarrow {(-5+4i)}^3+9{(-5+4i)}^2-(-5+4i)+4$

$\Rightarrow (-125+64i^3-3\times 5\times 4i(-5+4i))+9(25+16i^2-2\times 5\times 4i)+5-4i+4$

$\Rightarrow (-125-64i-60i(-5+4i))+9(25-16-40i)+9-4i$

$\Rightarrow (-125-64i+300i-240i^2)+9(9-40i)+9-4i$

$\Rightarrow 115+236i+90-360i-4i$

$\Rightarrow 205-128i$

Hence, this is the answer.