Question

Express the following in the form of $a + ib$, $a,b \in R$, $i = \sqrt { - 1} $. State the values of $a$ and $b$.
$\left( {1 + 2i} \right)\left( { - 2 + i} \right)$

Answer 1

Hint: In this question, we are given an equation and we have been asked to write it in the form of $a = ib$ and also, we have to state the values of a and b. We will start by multiplying the brackets of the given equation just like we multiply the other normal equations $\left( {a + b} \right)\left( {c + d} \right)$. After multiplying, we will rearrange the resultant equation by grouping the like terms and performing the addition, subtraction on them. After that, we will compare the final equation with $a + ib$. This will give us the values of a and b.

Complete step-by-step solution:
We are given an equation $\left( {1 + 2i} \right)\left( { - 2 + i} \right)$ and we have to express it in the form of $a = ib$ and also, we have to state the values of $a$ and $b$.
We will start with multiplying the brackets,
$ \Rightarrow \left( {1 + 2i} \right)\left( { - 2 + i} \right)$
On multiplying we will get,
$ \Rightarrow \left( { - 2 + i - 4i + 2{i^2}} \right)$
We know that $i = \sqrt { - 1} $. Therefore, if we square both the sides, we will get ${i^2} = - 1$. Putting in the above equation,
$ \Rightarrow \left( { - 2 + i - 4i + 2\left( { - 1} \right)} \right)$
On rearranging and simplifying we will get,
$ \Rightarrow \left( {\left( { - 2 - 2} \right) + \left( {i - 4i} \right)} \right)$
Simplifying further,
$ \Rightarrow - 4 - 3i$
Now, we have to state the values of $a$ and $b$.
Comparing the equation $a + ib$, we will get, $a = - 4$ and $b = - 3$.

$\therefore $ $a = - 4$ and $b = - 3$.

Note: 1) The equations we dealt with in this question are called complex equations because it includes $'i'$. The $'i'$ is an imaginary number as it cannot be located on the number line.
2) The value of $'i'$ is $\sqrt { - 1} $. We have to understand the powers of $'i'$.
$ \Rightarrow {i^2} = - 1$,
$ \Rightarrow {i^3} = {i^2}*i = - i$,
And finally, ${i^4} = 1$. These help in solving problems related to complex equations.