Courses
Courses for Kids
Free study material
Offline Centres
More

# How do you write ${\left( {2 + 3i} \right)^2}$ in standard form?

Last updated date: 01st Mar 2024
Total views: 342k
Views today: 3.42k
Verified
342k+ views
Hint:We know that a complex number has two components. The first part is a real number and the second part is an imaginary part. For example, let us take $x$ is the real part and $y$is the imaginary part of a complex number. Then, this complex number can be written in the standard form as $x + yi$. Thus, we will convert the given complex number into this type of standard form.

Formula used:
${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$

Our first step is to expand the given complex number ${\left( {2 + 3i} \right)^2}$.We have the formula for expanding a perfect square ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$.For this question, we will take $a = 2$ and $b = 3i$.
${\left( {2 + 3i} \right)^2} \\ \Rightarrow {\left( 2 \right)^2} + \left( {2 \times 2 \times 3i} \right) + {\left( {3i} \right)^2} \\ \Rightarrow 4 + 12i + 9{i^2} \\$
We know that the value of ${i^2}$is $- 1$.
$4 + 12i + 9\left( { - 1} \right) \\ \Rightarrow 4 + 12i - 9 \\ \therefore - 5 + 12i$
Here, $- 5$ is the real part of this standard form of complex number and $12$ is the imaginary part of this standard form of complex number.
Hence, our answer is: ${\left( {2 + 3i} \right)^2}$is written in its standard form as $- 5 + 12i$.