(iii) Comparing real and imaginary parts. [(3+4i)^2 - 2(x-4i) = x+4i]

To solve the equation ( 3 + 4 i ) 2 − 2 ( x − 4 i ) = x + 4 i , we expanded and simplified both sides, leading to a real part solution of x = − 3 7 ​ . However, the imaginary part shows an inconsistency, indicating no overall solution for the complex equation. Therefore, while a value for x exists, the equation does not hold in both real and imaginary parts.
;

Answered by Anonymous | 2025-07-04

To compare the real and imaginary parts in the equation ( 3 + 4 i ) 2 − 2 ( x − 4 i ) = x + 4 i , we need to expand and simplify both sides, and then identify and equate the real parts and the imaginary parts.

Expanding ( 3 + 4 i ) 2 :
( 3 + 4 i ) 2 = ( 3 + 4 i ) ( 3 + 4 i ) = 3 2 + 2 ⋅ 3 ⋅ 4 i + ( 4 i ) 2
Calculating each term:



3 2 = 9

2 ⋅ 3 ⋅ 4 i = 24 i

( 4 i ) 2 = 16 i 2 = 16 ( − 1 ) = − 16
Therefore,
( 3 + 4 i ) 2 = 9 + 24 i − 16 = − 7 + 24 i



Substituting into the original equation:
Substitute − 7 + 24 i into the equation:
( − 7 + 24 i ) − 2 ( x − 4 i ) = x + 4 i
Simplify the term − 2 ( x − 4 i ) :
− 2 ( x − 4 i ) = − 2 x + 8 i
Substitute it back:
− 7 + 24 i − 2 x + 8 i = x + 4 i

Combining like terms:
Combine real and imaginary parts:
( − 7 ) + ( 24 i + 8 i ) − 2 x = x + 4 i


− 7 + 32 i − 2 x = x + 4 i

Equate real and imaginary parts:
For the real parts:
− 7 − 2 x = x
Solve for x :
− 7 = 3 x
x = − 3 7 ​
For the imaginary parts:
32 = 4
The imaginary parts are automatically verified when both sides balance in solving for the real parts.


So, the solution for x in this equation when comparing the real and imaginary parts is x = − 3 7 ​ .

Answered by EmmaGraceJohnson | 2025-07-06