Question
Answer
$$(5m^2+2n^2-2mn)^2-(5m^2+2n^2-2mn)^2=0$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(5m^2+2n^2-2mn)^2-(5m^2+2n^2-2mn)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}25m^4-20m^3n+24m^2n^2-8mn^3+4n^4-(25m^4-20m^3n+24m^2n^2-8mn^3+4n^4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}25m^4-20m^3n+24m^2n^2-8mn^3+4n^4-25m^4+20m^3n-24m^2n^2+8mn^3-4n^4 \xlongequal{ } \\[1 em] & \xlongequal{ } \cancel{25m^4} -\cancel{20m^3n}+ \cancel{24m^2n^2} -\cancel{8mn^3}+ \cancel{4n^4} -\cancel{25m^4}+ \cancel{20m^3n} -\cancel{24m^2n^2}+ \cancel{8mn^3} -\cancel{4n^4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}0\end{aligned} $$ | |
| 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+2n^2-2mn}\right) $ by each term in $ \left( 5m^2+2n^2-2mn\right) $. $$ \left( \color{blue}{5m^2+2n^2-2mn}\right) \cdot \left( 5m^2+2n^2-2mn\right) = \\ = 25m^4+10m^2n^2-10m^3n+10m^2n^2+4n^4-4mn^3-10m^3n-4mn^3+4m^2n^2 $$ |
| ③ | Combine like terms: $$ 25m^4+ \color{blue}{10m^2n^2} \color{red}{-10m^3n} + \color{green}{10m^2n^2} +4n^4 \color{orange}{-4mn^3} \color{red}{-10m^3n} \color{orange}{-4mn^3} + \color{green}{4m^2n^2} = \\ = 25m^4 \color{red}{-20m^3n} + \color{green}{24m^2n^2} \color{orange}{-8mn^3} +4n^4 $$ |
| ④ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 25m^4-20m^3n+24m^2n^2-8mn^3+4n^4 \right) = -25m^4+20m^3n-24m^2n^2+8mn^3-4n^4 $$ |
| ⑤ | Combine like terms: $$ \, \color{blue}{ \cancel{25m^4}} \, \, \color{green}{ -\cancel{20m^3n}} \,+ \, \color{blue}{ \cancel{24m^2n^2}} \, \, \color{green}{ -\cancel{8mn^3}} \,+ \, \color{blue}{ \cancel{4n^4}} \, \, \color{blue}{ -\cancel{25m^4}} \,+ \, \color{green}{ \cancel{20m^3n}} \, \, \color{blue}{ -\cancel{24m^2n^2}} \,+ \, \color{green}{ \cancel{8mn^3}} \, \, \color{blue}{ -\cancel{4n^4}} \, = 0 $$ |
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