Answer
$$(a+bi)(a-bi)=-b^2i^2+a^2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(a+bi)(a-bi)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}a^2-abi+abi-b^2i^2 \xlongequal{ } \\[1 em] & \xlongequal{ }a^2 -\cancel{abi}+ \cancel{abi}-b^2i^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}-b^2i^2+a^2\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{a+bi}\right) $ by each term in $ \left( a-bi\right) $. $$ \left( \color{blue}{a+bi}\right) \cdot \left( a-bi\right) = a^2 -\cancel{abi}+ \cancel{abi}-b^2i^2 $$ |
| ② | Combine like terms: $$ a^2 \, \color{blue}{ -\cancel{abi}} \,+ \, \color{blue}{ \cancel{abi}} \,-b^2i^2 = -b^2i^2+a^2 $$ |
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