Problem 19 - Algebra and Trigonometry

Question

Find each product and write the result in standard form. $$ (2+3 i)^{2} $$

Step-by-Step Solution

Answer

The result of the product in standard form is \(-5 + 12i\)

Step 1: Identify the components of the complex number

Firstly, recognize the real and imaginary part of the complex number. In this case, the real part, a, of the complex number is 2 and the imaginary part, b, is 3.

Step 2: Apply the formula

Now use the formula \((a+b)^{2}=a^{2}+2ab+b^{2}\) to find the square of the complex number. So, \( (2+3i)^{2}=2^{2}+2(2)(3i)+(3i)^{2}\).

Step 3: Calculate

Calculate the values as \(4 + 12i - 9\), where \(i^{2}\) is -1

Step 4: Simplify

Simplify the expression as \(-5 + 12i\) which is in standard form