Tap the blue circles to see an explanation.
$$ \begin{aligned}\frac{(a+b+c)^2-(a-b-c)^2}{4a}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{a^2+2ab+2ac+b^2+2bc+c^2-(1a^2-2ab-2ac+b^2+2bc+c^2)}{4a} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{a^2+2ab+2ac+b^2+2bc+c^2-a^2+2ab+2ac-b^2-2bc-c^2}{4a} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{4ab+4ac}{4a}\end{aligned} $$ | |
① | Multiply each term of $ \left( \color{blue}{a+b+c}\right) $ by each term in $ \left( a+b+c\right) $. $$ \left( \color{blue}{a+b+c}\right) \cdot \left( a+b+c\right) = a^2+ab+ac+ab+b^2+bc+ac+bc+c^2 $$ |
② | Combine like terms: $$ a^2+ \color{blue}{ab} + \color{red}{ac} + \color{blue}{ab} +b^2+ \color{green}{bc} + \color{red}{ac} + \color{green}{bc} +c^2 = \\ = a^2+ \color{blue}{2ab} + \color{red}{2ac} +b^2+ \color{green}{2bc} +c^2 $$ |
③ | Multiply each term of $ \left( \color{blue}{a-b-c}\right) $ by each term in $ \left( a-b-c\right) $. $$ \left( \color{blue}{a-b-c}\right) \cdot \left( a-b-c\right) = a^2-ab-ac-ab+b^2+bc-ac+bc+c^2 $$ |
④ | Combine like terms: $$ a^2 \color{blue}{-ab} \color{red}{-ac} \color{blue}{-ab} +b^2+ \color{green}{bc} \color{red}{-ac} + \color{green}{bc} +c^2 = \\ = a^2 \color{blue}{-2ab} \color{red}{-2ac} +b^2+ \color{green}{2bc} +c^2 $$ |
⑤ | Remove the parentheses by changing the sign of each term within them. $$ - \left( a^2-2ab-2ac+b^2+2bc+c^2 \right) = -a^2+2ab+2ac-b^2-2bc-c^2 $$ |
⑥ | $$ \, \color{blue}{ \cancel{a^2}} \,+ \color{green}{2ab} + \color{orange}{2ac} + \, \color{blue}{ \cancel{b^2}} \,+ \, \color{green}{ \cancel{2bc}} \,+ \, \color{blue}{ \cancel{c^2}} \, \, \color{blue}{ -\cancel{a^2}} \,+ \color{green}{2ab} + \color{orange}{2ac} \, \color{blue}{ -\cancel{b^2}} \, \, \color{green}{ -\cancel{2bc}} \, \, \color{blue}{ -\cancel{c^2}} \, = \color{green}{4ab} + \color{orange}{4ac} $$ |