Question
Answer
$$(-3a+b)^2+(-a+b-1)^2+(a+b-1)^2+(3a+b-2)^2=20a^2+4b^2-12a-8b+6$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-3a+b)^2+(-a+b-1)^2+(a+b-1)^2+(3a+b-2)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}9a^2-6ab+b^2+a^2-2ab+b^2+2a-2b+1+a^2+2ab+b^2-2a-2b+1+9a^2+6ab+b^2-12a-4b+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}10a^2-8ab+2b^2+2a-2b+1+a^2+2ab+b^2-2a-2b+1+9a^2+6ab+b^2-12a-4b+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}11a^2-6ab+3b^2-4b+2+9a^2+6ab+b^2-12a-4b+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}20a^2+4b^2-12a-8b+6\end{aligned} $$ | |
| ① | Find $ \left(-3a+b\right)^2 $ in two steps. S1: Change all signs inside bracket. S2: Apply formula $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 3a } $ and $ B = \color{red}{ b }$. $$ \begin{aligned}\left(-3a+b\right)^2& \xlongequal{ S1 } \left(3a-b\right)^2 \xlongequal{ S2 } \color{blue}{\left( 3a \right)^2} -2 \cdot 3a \cdot b + \color{red}{b^2} = \\[1 em] & = 9a^2-6ab+b^2\end{aligned} $$Multiply each term of $ \left( \color{blue}{-a+b-1}\right) $ by each term in $ \left( -a+b-1\right) $. $$ \left( \color{blue}{-a+b-1}\right) \cdot \left( -a+b-1\right) = a^2-ab+a-ab+b^2-b+a-b+1 $$ |
| ② | Combine like terms: $$ a^2 \color{blue}{-ab} + \color{red}{a} \color{blue}{-ab} +b^2 \color{green}{-b} + \color{red}{a} \color{green}{-b} +1 = a^2 \color{blue}{-2ab} +b^2+ \color{red}{2a} \color{green}{-2b} +1 $$Multiply each term of $ \left( \color{blue}{a+b-1}\right) $ by each term in $ \left( a+b-1\right) $. $$ \left( \color{blue}{a+b-1}\right) \cdot \left( a+b-1\right) = a^2+ab-a+ab+b^2-b-a-b+1 $$ |
| ③ | Combine like terms: $$ a^2+ \color{blue}{ab} \color{red}{-a} + \color{blue}{ab} +b^2 \color{green}{-b} \color{red}{-a} \color{green}{-b} +1 = a^2+ \color{blue}{2ab} +b^2 \color{red}{-2a} \color{green}{-2b} +1 $$Multiply each term of $ \left( \color{blue}{3a+b-2}\right) $ by each term in $ \left( 3a+b-2\right) $. $$ \left( \color{blue}{3a+b-2}\right) \cdot \left( 3a+b-2\right) = 9a^2+3ab-6a+3ab+b^2-2b-6a-2b+4 $$ |
| ④ | Combine like terms: $$ 9a^2+ \color{blue}{3ab} \color{red}{-6a} + \color{blue}{3ab} +b^2 \color{green}{-2b} \color{red}{-6a} \color{green}{-2b} +4 = 9a^2+ \color{blue}{6ab} +b^2 \color{red}{-12a} \color{green}{-4b} +4 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{9a^2} \color{red}{-6ab} + \color{green}{b^2} + \color{blue}{a^2} \color{red}{-2ab} + \color{green}{b^2} +2a-2b+1 = \color{blue}{10a^2} \color{red}{-8ab} + \color{green}{2b^2} +2a-2b+1 $$ |
| ⑥ | Combine like terms: $$ \color{blue}{10a^2} \color{red}{-8ab} + \color{green}{2b^2} + \, \color{orange}{ \cancel{2a}} \, \color{red}{-2b} + \color{green}{1} + \color{blue}{a^2} + \color{red}{2ab} + \color{green}{b^2} \, \color{orange}{ -\cancel{2a}} \, \color{red}{-2b} + \color{green}{1} = \\ = \color{blue}{11a^2} \color{red}{-6ab} + \color{green}{3b^2} \color{red}{-4b} + \color{green}{2} $$ |
| ⑦ | Combine like terms: $$ \color{blue}{11a^2} \, \color{red}{ -\cancel{6ab}} \,+ \color{orange}{3b^2} \color{blue}{-4b} + \color{red}{2} + \color{blue}{9a^2} + \, \color{red}{ \cancel{6ab}} \,+ \color{orange}{b^2} -12a \color{blue}{-4b} + \color{red}{4} = \\ = \color{blue}{20a^2} + \color{orange}{4b^2} -12a \color{blue}{-8b} + \color{red}{6} $$ |
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