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

Hint: 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.