Question
Answer
$$(rx+ry)^2-2ry(rx+ry)=r^2x^2-r^2y^2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(rx+ry)^2-2ry(rx+ry)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}r^2x^2+2r^2xy+r^2y^2-2ry(rx+ry) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}r^2x^2+2r^2xy+r^2y^2-(2r^2xy+2r^2y^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}r^2x^2+2r^2xy+r^2y^2-2r^2xy-2r^2y^2 \xlongequal{ } \\[1 em] & \xlongequal{ }r^2x^2+ \cancel{2r^2xy}+r^2y^2 -\cancel{2r^2xy}-2r^2y^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}r^2x^2-r^2y^2\end{aligned} $$ | |
| ① | Find $ \left(rx+ry\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ rx } $ and $ B = \color{red}{ ry }$. $$ \begin{aligned}\left(rx+ry\right)^2 = \color{blue}{\left( rx \right)^2} +2 \cdot rx \cdot ry + \color{red}{\left( ry \right)^2} = r^2x^2+2r^2xy+r^2y^2\end{aligned} $$ |
| ② | Multiply $ \color{blue}{2ry} $ by $ \left( rx+ry\right) $ $$ \color{blue}{2ry} \cdot \left( rx+ry\right) = 2r^2xy+2r^2y^2 $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 2r^2xy+2r^2y^2 \right) = -2r^2xy-2r^2y^2 $$ |
| ④ | Combine like terms: $$ r^2x^2+ \, \color{blue}{ \cancel{2r^2xy}} \,+ \color{green}{r^2y^2} \, \color{blue}{ -\cancel{2r^2xy}} \, \color{green}{-2r^2y^2} = r^2x^2 \color{green}{-r^2y^2} $$ |
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