Question

100 points PLEASE ANSWER ASAP thank you so much

Answer 1

LHS

• (a+bi)²
• (a+bi)(a+bi)
• a²+b²i²+2abi
• a²-b²+2abi

RHS

• (a+b)(a-b)+2abi
• a²-b²+2abi

Hence verified

Answer 2

Answer:

$\textsf{Apply the Perfect Square Formula}\quad (a + b)^2 = a^2 + 2ab + b^2:$

$\implies (a+bi)^2=a^2+2abi+(bi)^2$

$\textsf{Apply exponent rule} \quad (a\cdot b)^c=a^cb^c:$

$\implies a^2+2abi+b^2i^2$

$i=\sqrt{-1} \implies i^2=-1$

$\textsf{Apply imaginary number rule} \quad i^2=-1:$

$\implies a^2+2abi+b^2(-1)$

$\implies a^2+2abi-b^2$

$\textsf{Collect the real parts:}$

$\implies a^2-b^2+2abi$

$\textsf{Factor the real parts by applying the Difference of Two Squares Formula}\\x^2-y^2=(x+y)(x-y) :$

$\implies (a+b)(a-b)+2abi$