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Question: Find the value of \({{\left( {{z}^{2}}+5z \right)}^{2}}+z\left( z+5 \right)\), when \(z= \dfrac{-5+\...

Find the value of (z2+5z)2+z(z+5){{\left( {{z}^{2}}+5z \right)}^{2}}+z\left( z+5 \right), when z=−5+3i2z= \dfrac{-5+\sqrt{3}i}{2}, is:
(a) 41
(b) 42
(c) 43
(d) 45

Explanation

Solution

Hint: First we will rearrange the given equation and then we will find the value z2{{z}^{2}} , after that we have to put that value in the given equation and then we will try to convert it in the form of a + ib.

Complete step-by-step answer:
Let’s first find the value of (z2+5z)\left( {{z}^{2}}+5z \right) :
The formula for (a−b)2=a2+b2−2ab{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab , we are going to use this formula for calculating the value of z2{{z}^{2}}.
Another formula that we are going to use is (a+b)(a−b)=a2−b2\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}} and i2=−1{{i}^{2}}=-1 ,
After putting the value of z in the given equation we get,
(−5+3i2)2+5(−5+3i2) =(25−3−103i4)+(−50+103i4) =−284 =−7 \begin{aligned} & {{\left( \dfrac{-5+\sqrt{3}i}{2} \right)}^{2}}+5\left( \dfrac{-5+\sqrt{3}i}{2} \right) \\\ & =\left( \dfrac{25-3-10\sqrt{3}i}{4} \right)+\left( \dfrac{-50+10\sqrt{3}i}{4} \right) \\\ & =\dfrac{-28}{4} \\\ & =-7 \\\ \end{aligned}
The reason we first find the value of (z2+5z)\left( {{z}^{2}}+5z \right) because if we put (z2+5z)\left( {{z}^{2}}+5z \right) = x in (z2+5z)2+z(z+5){{\left( {{z}^{2}}+5z \right)}^{2}}+z\left( z+5 \right) we will get x2+x{{x}^{2}}+x ,
We have converted the given complex equation in terms z to a simple quadratic equation which has now become very easy to solve.
So, the only variable now is x so that’s why we have to first find the value of this x.
And the value of x is = -7.
Hence, putting the value of x in x2+x{{x}^{2}}+x we get,
⇒(−7)2+(−7) ⇒49−7 ⇒42 \begin{aligned} & \Rightarrow {{\left( -7 \right)}^{2}}+\left( -7 \right) \\\ & \Rightarrow 49-7 \\\ & \Rightarrow 42 \\\ \end{aligned}
Hence, the correct answer is 42 which is option (b).

Note: There is another approach to solve this question, we could have put the value of z in the given equation directly and we can try to solve it one by one but if we do that it’s going to take lot of time so, instead of that go with the method that have been given in the solution in which it has been converted into a simple quadratic equation.