Question
Answer
$$(a+b+c+d)(a-b+c-d)=a^2+2ac-b^2-2bd+c^2-d^2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(a+b+c+d)(a-b+c-d)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}a^2+2ac-b^2-2bd+c^2-d^2\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{a+b+c+d}\right) $ by each term in $ \left( a-b+c-d\right) $. $$ \left( \color{blue}{a+b+c+d}\right) \cdot \left( a-b+c-d\right) = \\ = a^2 -\cancel{ab}+ac -\cancel{ad}+ \cancel{ab}-b^2+ \cancel{bc}-bd+ac -\cancel{bc}+c^2 -\cancel{cd}+ \cancel{ad}-bd+ \cancel{cd}-d^2 $$ |
| ② | Combine like terms: $$ a^2 \, \color{blue}{ -\cancel{ab}} \,+ \color{green}{ac} \, \color{orange}{ -\cancel{ad}} \,+ \, \color{blue}{ \cancel{ab}} \,-b^2+ \, \color{red}{ \cancel{bc}} \, \color{orange}{-bd} + \color{green}{ac} \, \color{red}{ -\cancel{bc}} \,+c^2 \, \color{blue}{ -\cancel{cd}} \,+ \, \color{orange}{ \cancel{ad}} \, \color{orange}{-bd} + \, \color{blue}{ \cancel{cd}} \,-d^2 = a^2+ \color{green}{2ac} -b^2 \color{orange}{-2bd} +c^2-d^2 $$ |
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