Question

If $2x=3+5i$, then the value of $2x^3+2x^2-7x+72$ is

• A. $4$
• B. $-4$
• C. $8$
• D. $-8$

Answer 1

The correct option is A

$4$

Explanation for the correct option:

Step 1. Find the value of $2x^3+2x^2-7x+72$:

Given, $2x=3+5i$

$\Rightarrow$ $x=\frac{(3+5i)}{2}$

Now, $\begin{array}{rcl}{x}^3& =& \frac{(27+135i-225-125i)}{8}\end{array}$

$\begin{array}{rcl}& =& \frac{(-198+10i)}{8}\end{array}$

$x^2=\frac{(9-25+30i)}{4}$

$=\frac{(-16+30i)}{4}$

Step 2. Put the value of $x^3$, $x^2$ and $x$ in the given expression:

$2x^3+2x^2–7x+72=\frac{2(-198+10i)}{8}+\frac{2(-16+30i)}{4}–\frac{7(3+5i)}{2}+72$

$=\left(\frac{-99}{2}-8-\frac{21}{2}+72\right)+\left(\frac{10}{4}+15-\frac{35}{2}\right)i$

$=\frac{(-99-16-21+144)}{2}+\left[\frac{(10+60-70)}{4}\right]i$

$=\frac{8}{2}$

$=\mathbf{4}$

Hence, Option ‘A’ is Correct.