Question

Express $(4-5i)(-5+5i)$ in the form $a+bi$, where $a$ and $b$ are integers and $i^2 = -1.$

a) $-5+45i$
b) $-25+5i$
c) $25-5i$
d) $-25-5i$

Answer 1

The multiplication of complex numbers (4-5i)(-5+5i) results in 5 + 45i, after performing the necessary operations with real and imaginary parts. The final result is in the form a+bi, where a and b are integers.

Step-by-step explanation:

To express (4-5i)(-5+5i) in the form a+bi, where a and b are integers and i is the imaginary unit with the property that i2 = -1, we need to multiply the two complex numbers.

Here is the step-by-step calculation:

1. Multiply the real part of the first complex number by the real part of the second: 4 x (-5) = -20.
2. Multiply the imaginary part of the first complex number by the imaginary part of the second (remembering to use the property i2 = -1): (-5i)(5i) = -25i2 = -25(-1) = 25.
3. Multiply the real part of the first number by the imaginary part of the second and vice versa, then add the two products: 4(5i) + (-5)(-5i) = 20i + 25i = 45i.
4. Combine the results from steps 1 and 2 to get the real part, and the result from step 3 for the imaginary part: -20 + 25 + 45i = 5 + 45i.

Therefore, (4-5i)(-5+5i) = 5 + 45i, which means the correct answer is option a) -5 + 45i.