Question

let a be a real number such that the following product is also equal to a real number.
(2 + 3i)(a - 3i)
what is the value of a?
a =

Explanation:

Step1: Expand the product

Using the distributive property (FOIL method) for complex numbers: \((2 + 3i)(a - 3i)=2a-6i + 3ai-9i^{2}\). Since \(i^{2}=- 1\), this becomes \(2a-6i + 3ai + 9=(2a + 9)+(3a - 6)i\).

Step2: Set the imaginary part to zero

For the product to be a real number, the imaginary part must be zero. So we set \(3a-6 = 0\).

Step3: Solve for \(a\)

Add 6 to both sides: \(3a=6\). Divide both sides by 3: \(a = 2\).

Answer:

\(a = 2\)