Question
Answer
$$(x+y-3z)^2+(y+z-3x)^2+(z+x-3y)^2=11x^2-10xy-10xz+11y^2-10yz+11z^2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+y-3z)^2+(y+z-3x)^2+(z+x-3y)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^2+2xy-6xz+y^2-6yz+9z^2+9x^2-6xy-6xz+y^2+2yz+z^2+x^2-6xy+2xz+9y^2-6yz+z^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}10x^2-4xy-12xz+2y^2-4yz+10z^2+x^2-6xy+2xz+9y^2-6yz+z^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}11x^2-10xy-10xz+11y^2-10yz+11z^2\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x+y-3z}\right) $ by each term in $ \left( x+y-3z\right) $. $$ \left( \color{blue}{x+y-3z}\right) \cdot \left( x+y-3z\right) = x^2+xy-3xz+xy+y^2-3yz-3xz-3yz+9z^2 $$ |
| ② | Combine like terms: $$ x^2+ \color{blue}{xy} \color{red}{-3xz} + \color{blue}{xy} +y^2 \color{green}{-3yz} \color{red}{-3xz} \color{green}{-3yz} +9z^2 = \\ = x^2+ \color{blue}{2xy} \color{red}{-6xz} +y^2 \color{green}{-6yz} +9z^2 $$Multiply each term of $ \left( \color{blue}{y+z-3x}\right) $ by each term in $ \left( y+z-3x\right) $. $$ \left( \color{blue}{y+z-3x}\right) \cdot \left( y+z-3x\right) = y^2+yz-3xy+yz+z^2-3xz-3xy-3xz+9x^2 $$ |
| ③ | Combine like terms: $$ y^2+ \color{blue}{yz} \color{red}{-3xy} + \color{blue}{yz} +z^2 \color{green}{-3xz} \color{red}{-3xy} \color{green}{-3xz} +9x^2 = \\ = 9x^2 \color{red}{-6xy} \color{green}{-6xz} +y^2+ \color{blue}{2yz} +z^2 $$Multiply each term of $ \left( \color{blue}{z+x-3y}\right) $ by each term in $ \left( z+x-3y\right) $. $$ \left( \color{blue}{z+x-3y}\right) \cdot \left( z+x-3y\right) = z^2+xz-3yz+xz+x^2-3xy-3yz-3xy+9y^2 $$ |
| ④ | Combine like terms: $$ z^2+ \color{blue}{xz} \color{red}{-3yz} + \color{blue}{xz} +x^2 \color{green}{-3xy} \color{red}{-3yz} \color{green}{-3xy} +9y^2 = \\ = x^2 \color{green}{-6xy} + \color{blue}{2xz} +9y^2 \color{red}{-6yz} +z^2 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{x^2} + \color{red}{2xy} \color{green}{-6xz} + \color{orange}{y^2} \color{blue}{-6yz} + \color{red}{9z^2} + \color{blue}{9x^2} \color{red}{-6xy} \color{green}{-6xz} + \color{orange}{y^2} + \color{blue}{2yz} + \color{red}{z^2} = \\ = \color{blue}{10x^2} \color{red}{-4xy} \color{green}{-12xz} + \color{orange}{2y^2} \color{blue}{-4yz} + \color{red}{10z^2} $$ |
| ⑥ | Combine like terms: $$ \color{blue}{10x^2} \color{red}{-4xy} \color{green}{-12xz} + \color{orange}{2y^2} \color{blue}{-4yz} + \color{red}{10z^2} + \color{blue}{x^2} \color{red}{-6xy} + \color{green}{2xz} + \color{orange}{9y^2} \color{blue}{-6yz} + \color{red}{z^2} = \\ = \color{blue}{11x^2} \color{red}{-10xy} \color{green}{-10xz} + \color{orange}{11y^2} \color{blue}{-10yz} + \color{red}{11z^2} $$ |
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