Final answer:
To multiply (2i+3)(4i-5), apply the distributive property, multiply corresponding terms, and combine like terms to get the final answer, which is -23 + 2i.
Step-by-step explanation:
To multiply two complex numbers such as (2i+3)(4i-5), we need to apply the distributive property (also known as the FOIL method in this context), where we multiply each term of the first complex number by each term of the second complex number. Similar to multiplying binomials, this process involves multiplying the real parts, the imaginary parts, and then combining like terms.
Here's how it works step by step:
- First, multiply the imaginary parts: (2i)(4i) = 8i^2. Since i^2 = -1, this gives us -8.
- Second, multiply the real part of the first by the imaginary part of the second: (3)(4i) = 12i.
- Third, multiply the imaginary part of the first by the real part of the second: (2i)(-5) = -10i.
- Lastly, multiply the real parts: (3)(-5) = -15.
Now, combine the results: -8 (from the imaginary parts' multiplication) and -15 (from the real parts' multiplication) give us the real part of the result, and 12i and -10i combined give us the imaginary part.
The final answer is -8 - 15 + (12i - 10i), which simplifies to -23 + 2i.
Remembering the rules of multiplication, such as when multiplying like signs the result is positive, and when multiplying unlike signs the result is negative, can help avoid mistakes.