Question

: If $2x = 3 + 5i$, then the value of $2{x^3} + 2{x^2} - 7x + 72$ is
1 $4$
2 $- 4$
3 $8$
4 $- 8$

Answer 1

To find the value of $2{x^3} + 2{x^2} - 7x + 72$, we need to find the value of $x$ from the given expression $2x = 3 + 5i$, then from the obtained value of $x$; find ${x^2}$ and ${x^3}$ respectively. Then substitute the obtained value of $x$, ${x^2}$ and ${x^3}$ in the given expression $2{x^3} + 2{x^2} - 7x + 72$.

Complete step by step answer:
Let us write the given data:
$2x = 3 + 5i$
In which the equation can be written in terms of $x$as:
$x = \dfrac{{3 + 5i}}{2}$
Then with respect to $x$, ${x^3}$ becomes
${x^3} = {\left( {\dfrac{{3 + 5i}}{2}} \right)^3}$
Expanding the cube root terms, we get:
$\Rightarrow {x^3} = \left( {\dfrac{{27 + 135i - 225 - 125i}}{8}} \right)$
$\Rightarrow {x^3} = \dfrac{{\left( { - 198 + 10i} \right)}}{8}$
And now to obtain the value of ${x^2}$, we have:
$\Rightarrow {x^2} = \dfrac{{\left( {9 - 25 + 30i} \right)}}{4}$
$\Rightarrow {x^2} = \dfrac{{\left( { - 16 + 30i} \right)}}{4}$
Hence, to find the value of $2{x^3} + 2{x^2} - 7x + 72$, we need to substitute all the obtained values of $x$, ${x^2}$and${x^3}$i.e.,
$2{x^3} + 2{x^2} - 7x + 72 = 2\dfrac{{\left( { - 198 + 10i} \right)}}{8} + 2\dfrac{{\left( { - 16 + 30i} \right)}}{4} - 7\dfrac{{\left( {3 + 5i} \right)}}{2} + 72$
Evaluate each term, as:
$= \left( {\dfrac{{ - 99}}{2} - 8 - \dfrac{{21}}{2} + 72} \right) + \left[ {\left( {\dfrac{{10}}{4} + 15} \right)\dfrac{{ - 35}}{2}} \right]i$
Simplifying the terms, we have:
$= \left( {\dfrac{{ - 99 - 16 - 21 + 144}}{2}} \right) + \left( {\dfrac{{10 + 60 - 70}}{4}} \right)i$
Evaluating the numerator terms, we get:
$= \left( {\dfrac{8}{2}} \right) + \left( {\dfrac{0}{4}} \right)i$
$= \dfrac{8}{2}$
Hence, we get:
$= 4$
$\therefore 2{x^3} + 2{x^2} - 7x + 72 = 4$

So, the correct answer is “Option 1”.

Note: The key point to note is that, the given expression is not quadratic hence, while finding the value of $x$ from the given expression we must also find the value of $x$, such that find ${x^2}$ and ${x^3}$ we need to substitute in the given asked expression.