a^2 + b^2 factorization

沒有什麼矛盾
(a+bi)(a-bi)
=a²-abi+abi-(bi)²
=a²-(bi)²
由於i²=-1
-(bi)²=-(b)²(i)²=-b²(-1)=b²
所以a²-(bi)²=a²+b²

點解最後 - (bi)^2= +b^2
- (bi)^2
= - (b^2 i^2)
= - [b^2(-1)]
=+b^2

關於 - (bi)^2= b^2 :
因為 - (bi)^2= - ( (b)^2 ) ( (i)^2 ) = - (b^2) (-1) = b^2

呢個問題涉及到complex no. system,
除o左要知道 i^2 = -1 之外仲要識d complex no. o既property
例如(a+bi)(a-bi)=a(a-bi) + bi(a-bi)就已經係complex no. o既definition之一

p.s. a^2+b^2=(a+b)^2-2ab 都係常用o既工具之一，不過呢個係completing square, 唔係factorization

=(a+bi)(a-bi)
=a²-(bi)²
=a²-(-b²)
=a²+b²

(bi) ^ 2 = (b ^ 2)(i ^ 2) = (b ^ 2) (-1) = -b ^ 2

Therefore, -(bi) ^ 2 = -(-b ^ 2) = +b ^ 2