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 given expression:

Given, $2x=3+5i$

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

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

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

$\begin{array}{rcl}{x}^2& =& \frac{(9-25+30i)}{4}\\ & =& \frac{(30i-16)}{4}\end{array}$

Step 2. Substitute $x^3,x^2$ and $x$ in $2x^3+2x^2-7x+72$

$\begin{array}{rcl}& =& 2\left[\frac{(-198+10i)}{8}\right]+2\left[\frac{(30i-16)}{4}\right]-7\left[\frac{(3+5i)}{2}\right]+72\\ & =& \left(\frac{-99}{2}-8-\frac{21}{2}+72\right)+\left(\frac{10}{4}+15-\frac{35}{2}\right)i\\ & =& \left[\frac{(-99-16-21+144)}{2}\right]+\left[\frac{(10+60-70)}{4}\right]i\\ & =& \frac{(-136+144)}{2}\\ & =& \frac{8}{2}\\ & =& \mathbf{4}\end{array}$

Hence, option ‘A’ is Correct.