Question

Check whether it is true or not, if ${z^2} = - 3 + 4i$ then $z = \pm \left( {1 + 2i} \right)$?

Answer 1

Hint: Here, we will proceed by converting the RHS of the given equation i.e., ${z^2} = - 3 + 4i$ in such a ways that the RHS is in the form of ${a^2} + {b^2} + 2\left( a \right)\left( b \right)$ so that we can use the formula ${a^2} + {b^2} + 2\left( a \right)\left( b \right) = {\left( {a + b} \right)^2}$.

Complete step-by-step answer:
Given, ${z^2} = - 3 + 4i$
We can rewrite the above equation by replacing -3 with (-4+1) as given under
$
  {z^2} = - 4 + 1 + 4i \\
   \Rightarrow {z^2} = \left( { - 1} \right)4 + 1 + 4i \\
 $
As we know that ${i^2} = - 1$, the above equation becomes
$
   \Rightarrow {z^2} = \left( {{i^2}} \right)4 + 1 + 4i \\
   \Rightarrow {z^2} = 4{i^2} + 1 + 4i \\
 $
We can rewrite the above equation by replacing $4{i^2}$ by ${\left( {2i} \right)^2}$, we have
$ \Rightarrow {z^2} = {\left( {2i} \right)^2} + 1 + 4i$
We can rewrite the above equation by replacing 1 by ${\left( 1 \right)^2}$ and $4i$ by $2\left( {2i} \right)\left( 1 \right)$ , we have
$ \Rightarrow {z^2} = {\left( {2i} \right)^2} + {\left( 1 \right)^2} + 2\left( {2i} \right)\left( 1 \right)$
Using the formula ${a^2} + {b^2} + 2\left( a \right)\left( b \right) = {\left( {a + b} \right)^2}$, we get
$
   \Rightarrow {z^2} = {\left( {2i + 1} \right)^2} \\
   \Rightarrow {z^2} = {\left( {1 + 2i} \right)^2} \\
 $
By taking square root on both sides, we get
$
   \Rightarrow z = \pm \sqrt {{{\left( {1 + 2i} \right)}^2}} \\
   \Rightarrow z = \pm \left( {1 + 2i} \right) \\
 $
Therefore, if ${z^2} = - 3 + 4i$ then $z = \pm \left( {1 + 2i} \right)$ is true.

Note: In this particular problem, the most important step is $z = \pm \sqrt {{{\left( {1 + 2i} \right)}^2}} $ where we can clearly see that after taking the square root on both the sides of the equation ${z^2} = {\left( {1 + 2i} \right)^2}$, we have considered both positive as well as negative signs. So, the result is $z = \pm \left( {1 + 2i} \right)$ which means that the complex number z has two values which are $z = 1 + 2i$ and $z = - 1 - 2i$.