a + bi = (2 + 3i)(2 - 3i), the value of b is
Understand the Problem
The question is asking us to calculate the value of 'b' in the complex number 'a + bi', where 'a' and 'b' are real components derived from the product of two complex numbers, (2 + 3i) and (2 - 3i). We will solve this by first calculating the product of the two complex numbers and then identifying the imaginary component which corresponds to 'b'.
Answer
The value of 'b' is $0$.
Answer for screen readers
The value of 'b' is $0$.
Steps to Solve
- Calculate the product of the complex numbers
We need to multiply the two complex numbers $(2 + 3i)$ and $(2 - 3i)$. We use the distributive property (FOIL method):
$$(2 + 3i)(2 - 3i) = 2 \cdot 2 + 2 \cdot (-3i) + 3i \cdot 2 + 3i \cdot (-3i)$$
This simplifies to:
$$= 4 - 6i + 6i - 9(-1)$$
- Simplify the expression
The imaginary parts $-6i + 6i$ cancel each other out, and we simplify the expression:
$$= 4 + 9 = 13$$
Thus, the product of the two complex numbers is $13 + 0i$.
- Identify the imaginary component
In the resulting complex number $13 + 0i$, the component 'b' corresponds to the coefficient of $i$.
Therefore, $b = 0$.
The value of 'b' is $0$.
More Information
The resulting value of 'b' indicates that there is no imaginary part in the product of the two complex numbers, meaning the product is a real number.
Tips
- Failing to apply the distributive property correctly when multiplying the complex numbers.
- Confusing the real part and the imaginary part after simplification.
