Multiply $(6-5 i)^2$.
A) $11+60 i$
B) $11-60 i$
C) $36+25 i$
D) $36-25 i$
Answer (1)
Expand the expression: ( 6 − 5 i ) 2 = ( 6 − 5 i ) ( 6 − 5 i ) .
Apply the distributive property: ( 6 ) ( 6 ) + ( 6 ) ( − 5 i ) + ( − 5 i ) ( 6 ) + ( − 5 i ) ( − 5 i ) = 36 − 30 i − 30 i + 25 i 2 .
Substitute i 2 = − 1 : 36 − 60 i − 25 .
Simplify: 11 − 60 i . The final answer is 11 − 60 i .
Explanation
Understanding the Problem We are asked to multiply the complex number ( 6 − 5 i ) by itself, which means we need to calculate ( 6 − 5 i ) 2 . The result should be in the form a + bi , where a and b are real numbers. We need to expand the expression and simplify it using the fact that i 2 = − 1 .
Expanding the Expression We will expand the expression ( 6 − 5 i ) 2 as ( 6 − 5 i ) ( 6 − 5 i ) . Then, we apply the distributive property (also known as the FOIL method) to multiply the two complex numbers:
( 6 − 5 i ) ( 6 − 5 i ) = ( 6 ) ( 6 ) + ( 6 ) ( − 5 i ) + ( − 5 i ) ( 6 ) + ( − 5 i ) ( − 5 i )
Simplifying the terms Now, we simplify each term:
36 − 30 i − 30 i + 25 i 2
Substituting i^2 with -1 Next, we replace i 2 with − 1 :
36 − 30 i − 30 i + 25 ( − 1 ) = 36 − 60 i − 25
Combining real terms Finally, we combine the real terms:
36 − 25 − 60 i = 11 − 60 i
Final Answer So, ( 6 − 5 i ) 2 = 11 − 60 i . Therefore, the correct answer is B) 11 − 60 i .
Examples
Complex numbers might seem abstract, but they're incredibly useful in fields like electrical engineering. Imagine you're designing a circuit. The flow of electricity isn't always straightforward; it can have both a 'real' component (resistance) and an 'imaginary' component (reactance, which is due to capacitors and inductors). By using complex numbers, engineers can easily analyze and predict the behavior of AC circuits. For example, the impedance (a generalization of resistance) in an AC circuit is often expressed as a complex number, allowing for straightforward calculations of voltage and current using Ohm's Law.
