Question
Answer
$$(5m^2+2n^2-2mn)^2-(5m^2-n^2+4mn)^2=-60m^3n+18m^2n^2+3n^4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(5m^2+2n^2-2mn)^2-(5m^2-n^2+4mn)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}25m^4-20m^3n+24m^2n^2-8mn^3+4n^4-(25m^4+40m^3n+6m^2n^2-8mn^3+n^4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}25m^4-20m^3n+24m^2n^2-8mn^3+4n^4-25m^4-40m^3n-6m^2n^2+8mn^3-n^4 \xlongequal{ } \\[1 em] & \xlongequal{ } \cancel{25m^4}-20m^3n+24m^2n^2 -\cancel{8mn^3}+4n^4 -\cancel{25m^4}-40m^3n-6m^2n^2+ \cancel{8mn^3}-n^4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-60m^3n+18m^2n^2+3n^4\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{5m^2+2n^2-2mn}\right) $ by each term in $ \left( 5m^2+2n^2-2mn\right) $. $$ \left( \color{blue}{5m^2+2n^2-2mn}\right) \cdot \left( 5m^2+2n^2-2mn\right) = \\ = 25m^4+10m^2n^2-10m^3n+10m^2n^2+4n^4-4mn^3-10m^3n-4mn^3+4m^2n^2 $$ |
| ② | Combine like terms: $$ 25m^4+ \color{blue}{10m^2n^2} \color{red}{-10m^3n} + \color{green}{10m^2n^2} +4n^4 \color{orange}{-4mn^3} \color{red}{-10m^3n} \color{orange}{-4mn^3} + \color{green}{4m^2n^2} = \\ = 25m^4 \color{red}{-20m^3n} + \color{green}{24m^2n^2} \color{orange}{-8mn^3} +4n^4 $$Multiply each term of $ \left( \color{blue}{5m^2-n^2+4mn}\right) $ by each term in $ \left( 5m^2-n^2+4mn\right) $. $$ \left( \color{blue}{5m^2-n^2+4mn}\right) \cdot \left( 5m^2-n^2+4mn\right) = \\ = 25m^4-5m^2n^2+20m^3n-5m^2n^2+n^4-4mn^3+20m^3n-4mn^3+16m^2n^2 $$ |
| ③ | Combine like terms: $$ 25m^4 \color{blue}{-5m^2n^2} + \color{red}{20m^3n} \color{green}{-5m^2n^2} +n^4 \color{orange}{-4mn^3} + \color{red}{20m^3n} \color{orange}{-4mn^3} + \color{green}{16m^2n^2} = \\ = 25m^4+ \color{red}{40m^3n} + \color{green}{6m^2n^2} \color{orange}{-8mn^3} +n^4 $$ |
| ④ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 25m^4+40m^3n+6m^2n^2-8mn^3+n^4 \right) = -25m^4-40m^3n-6m^2n^2+8mn^3-n^4 $$ |
| ⑤ | Combine like terms: $$ \, \color{blue}{ \cancel{25m^4}} \, \color{green}{-20m^3n} + \color{orange}{24m^2n^2} \, \color{blue}{ -\cancel{8mn^3}} \,+ \color{green}{4n^4} \, \color{blue}{ -\cancel{25m^4}} \, \color{green}{-40m^3n} \color{orange}{-6m^2n^2} + \, \color{blue}{ \cancel{8mn^3}} \, \color{green}{-n^4} = \\ = \color{green}{-60m^3n} + \color{orange}{18m^2n^2} + \color{green}{3n^4} $$ |
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