Question

How do you simplify (2 + 6i).(2 – 9i) ?

Answer 1

Hint: We will first use the fact that $(a + b).(c + d) = a.(c + d) + b.(c + d)$. Then we will further use the distributive property to simplify it and further and then combine the like terms together.

Complete step by step solution:
We are given that we need to simplify $(2 + 6i).(2 – 9i).$
Since we know that we have a fact given by the following expression:-
$ \Rightarrow $ $(a + b).(c + d) = a.(c + d) + b.(c + d)$
Replacing a by 2, b by 6i, c by 2 and d by – 9i, we will then obtain the following expression as:-
$ \Rightarrow (2 + 6i).(2 – 9i) = 2 (2 – 9i) + 6i (2 – 9i)$ …………………(1)
Now, we will use distributive property on $2 (2 – 9i)$ so that we will obtain the following equation:-
$ \Rightarrow 2 (2 – 9i) = 2 (2) + 2 (- 9i)$
Simplifying the right hand side of the above equation by calculating the required, we will then obtain the following expression as:-
$ \Rightarrow 2 (2 – 9i) = 4 – 18i $ …………….(2)
Now, we will use distributive property on $6i (2 – 9i)$ so that we will obtain the following equation:-
$ \Rightarrow 6i (2 – 9i) = 6i (2) + 6i (- 9i)$
Simplifying the right hand side of the above equation by calculating the required, we will then obtain the following expression as:-
$ \Rightarrow 6i(2 - 9i) = 12i - 54{i^2}$
Since, we know that ${i^2} = - 1$, therefore, we have the following equation as:-
$ \Rightarrow 6i(2 - 9i) = 12i - 54( - 1)$
Simplifying the right hand side of the above equation by calculating the required, we will then obtain the following expression as:-
$ \Rightarrow 6i (2 – 9i) = 12i + 54$ ………………(3)
Putting the equation number 2 and equation number 3 in equation number 1, we will then obtain the following equation as:-
$ \Rightarrow (2 + 6i).(2 – 9i) = 4 – 18i + 12i + 54$
Simplifying the right hand side of the above equation by simplifying the calculations (by clubbing the right hand side) further, we will then obtain the following equation as:-
$ \Rightarrow (2 + 6i).(2 – 9i) = 58 – 6i$
Thus, we have the required answer.

Note: The students must note that the distributive property as we mentioned in the above solution states that:
For any numbers a, b and c, we have the following equation:-
$ \Rightarrow a (b + c) = ab + ac$
This is true for all a, b and c (not only real numbers but also complex numbers)
The students must also note that $i = \sqrt { - 1} $.
Squaring both the sides of the above equation, we will then obtain the following equation with us:-
$ \Rightarrow {i^2} = \sqrt { - 1} \times \sqrt { - 1} $
$ \Rightarrow {i^2} = - 1$