Question
Answer
$$2(27u-16v)^2-3(9u^2-8v^2)=1431u^2-1728uv+536v^2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}2(27u-16v)^2-3(9u^2-8v^2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}2(729u^2-864uv+256v^2)-3(9u^2-8v^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}1458u^2-1728uv+512v^2-(27u^2-24v^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}1458u^2-1728uv+512v^2-27u^2+24v^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}1431u^2-1728uv+536v^2\end{aligned} $$ | |
| ① | Find $ \left(27u-16v\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 27u } $ and $ B = \color{red}{ 16v }$. $$ \begin{aligned}\left(27u-16v\right)^2 = \color{blue}{\left( 27u \right)^2} -2 \cdot 27u \cdot 16v + \color{red}{\left( 16v \right)^2} = 729u^2-864uv+256v^2\end{aligned} $$ |
| ② | Multiply $ \color{blue}{2} $ by $ \left( 729u^2-864uv+256v^2\right) $ $$ \color{blue}{2} \cdot \left( 729u^2-864uv+256v^2\right) = 1458u^2-1728uv+512v^2 $$Multiply $ \color{blue}{3} $ by $ \left( 9u^2-8v^2\right) $ $$ \color{blue}{3} \cdot \left( 9u^2-8v^2\right) = 27u^2-24v^2 $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 27u^2-24v^2 \right) = -27u^2+24v^2 $$ |
| ④ | Combine like terms: $$ \color{blue}{1458u^2} -1728uv+ \color{red}{512v^2} \color{blue}{-27u^2} + \color{red}{24v^2} = \color{blue}{1431u^2} -1728uv+ \color{red}{536v^2} $$ |
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