What values of [tex]$c$[/tex] and [tex]$d$[/tex] would make the following expression represent a real number?
[tex]$i(2+3 i)(c+d i)$[/tex]
A. [tex]$c=2, d=3$[/tex]
B. [tex]$c=-3, d=-2$[/tex]
C. [tex]$c=3, d=-2$[/tex]
D. [tex]$c=-2, d=3$[/tex]
Answer (2)
Expand the expression i ( 2 + 3 i ) ( c + d i ) to get − ( 2 d + 3 c ) + ( 2 c − 3 d ) i .
Set the imaginary part equal to zero: 2 c − 3 d = 0 .
Check the given options to see which one satisfies the equation 2 c − 3 d = 0 .
The correct option is c = − 3 , d = − 2 , which makes the expression a real number: c = − 3 , d = − 2 .
Explanation
Understanding the Problem We are given the expression i ( 2 + 3 i ) ( c + d i ) and we want to find values of c and d such that the expression is a real number. This means that the imaginary part of the expression must be equal to 0.
Expanding the Expression Let's expand the expression:
i ( 2 + 3 i ) ( c + d i ) = i ( 2 c + 2 d i + 3 c i + 3 d i 2 ) = i ( 2 c + 2 d i + 3 c i − 3 d ) = i (( 2 c − 3 d ) + ( 2 d + 3 c ) i ) = ( 2 c − 3 d ) i + ( 2 d + 3 c ) i 2 = ( 2 c − 3 d ) i − ( 2 d + 3 c )
So, i ( 2 + 3 i ) ( c + d i ) = − ( 2 d + 3 c ) + ( 2 c − 3 d ) i .
Setting the Imaginary Part to Zero For the expression to be a real number, the imaginary part must be zero. Therefore, we must have:
2 c − 3 d = 0
Solving for c in terms of d Now we can solve for c in terms of d :
2 c = 3 d
c = 2 3 d
Checking the Options Now we will check the given options to see which one satisfies the equation 2 c − 3 d = 0 :
Option 1: c = 2 , d = 3 2 ( 2 ) − 3 ( 3 ) = 4 − 9 = − 5 = 0
Option 2: c = − 3 , d = − 2 2 ( − 3 ) − 3 ( − 2 ) = − 6 + 6 = 0
Option 3: c = 3 , d = − 2 2 ( 3 ) − 3 ( − 2 ) = 6 + 6 = 12 = 0
Option 4: c = − 2 , d = 3 2 ( − 2 ) − 3 ( 3 ) = − 4 − 9 = − 13 = 0
Only option 2 satisfies the condition 2 c − 3 d = 0 .
Final Answer Therefore, the values of c and d that make the expression a real number are c = − 3 and d = − 2 .
Examples
Complex numbers are used in electrical engineering to represent alternating currents. The impedance of a circuit, which is the opposition to the flow of current, is a complex number. By manipulating complex numbers, engineers can analyze and design circuits. For example, the expression i ( 2 + 3 i ) ( c + d i ) could represent a simplified model of a circuit, and finding values of c and d that make the expression real could correspond to finding conditions for resonance in the circuit.
The values of c and d that make the expression i ( 2 + 3 i ) ( c + d i ) a real number are c = − 3 and d = − 2 . This is derived by setting the imaginary part of the expression equal to zero. The correct answer is Option B.
;
