Question
Answer
$$(-4a+b)^2+(-3a+b-2)^2+(3a+b-2)^2+(4a+b-4)^2=50a^2+4b^2-32a-16b+24$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-4a+b)^2+(-3a+b-2)^2+(3a+b-2)^2+(4a+b-4)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}16a^2-8ab+b^2+9a^2-6ab+b^2+12a-4b+4+9a^2+6ab+b^2-12a-4b+4+16a^2+8ab+b^2-32a-8b+16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}25a^2-14ab+2b^2+12a-4b+4+9a^2+6ab+b^2-12a-4b+4+16a^2+8ab+b^2-32a-8b+16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}34a^2-8ab+3b^2-8b+8+16a^2+8ab+b^2-32a-8b+16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}50a^2+4b^2-32a-16b+24\end{aligned} $$ | |
| ① | Find $ \left(-4a+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}{ 4a } $ and $ B = \color{red}{ b }$. $$ \begin{aligned}\left(-4a+b\right)^2& \xlongequal{ S1 } \left(4a-b\right)^2 \xlongequal{ S2 } \color{blue}{\left( 4a \right)^2} -2 \cdot 4a \cdot b + \color{red}{b^2} = \\[1 em] & = 16a^2-8ab+b^2\end{aligned} $$Multiply each term of $ \left( \color{blue}{-3a+b-2}\right) $ by each term in $ \left( -3a+b-2\right) $. $$ \left( \color{blue}{-3a+b-2}\right) \cdot \left( -3a+b-2\right) = 9a^2-3ab+6a-3ab+b^2-2b+6a-2b+4 $$ |
| ② | Combine like terms: $$ 9a^2 \color{blue}{-3ab} + \color{red}{6a} \color{blue}{-3ab} +b^2 \color{green}{-2b} + \color{red}{6a} \color{green}{-2b} +4 = 9a^2 \color{blue}{-6ab} +b^2+ \color{red}{12a} \color{green}{-4b} +4 $$Multiply each term of $ \left( \color{blue}{3a+b-2}\right) $ by each term in $ \left( 3a+b-2\right) $. $$ \left( \color{blue}{3a+b-2}\right) \cdot \left( 3a+b-2\right) = 9a^2+3ab-6a+3ab+b^2-2b-6a-2b+4 $$ |
| ③ | Combine like terms: $$ 9a^2+ \color{blue}{3ab} \color{red}{-6a} + \color{blue}{3ab} +b^2 \color{green}{-2b} \color{red}{-6a} \color{green}{-2b} +4 = 9a^2+ \color{blue}{6ab} +b^2 \color{red}{-12a} \color{green}{-4b} +4 $$Multiply each term of $ \left( \color{blue}{4a+b-4}\right) $ by each term in $ \left( 4a+b-4\right) $. $$ \left( \color{blue}{4a+b-4}\right) \cdot \left( 4a+b-4\right) = 16a^2+4ab-16a+4ab+b^2-4b-16a-4b+16 $$ |
| ④ | Combine like terms: $$ 16a^2+ \color{blue}{4ab} \color{red}{-16a} + \color{blue}{4ab} +b^2 \color{green}{-4b} \color{red}{-16a} \color{green}{-4b} +16 = \\ = 16a^2+ \color{blue}{8ab} +b^2 \color{red}{-32a} \color{green}{-8b} +16 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{16a^2} \color{red}{-8ab} + \color{green}{b^2} + \color{blue}{9a^2} \color{red}{-6ab} + \color{green}{b^2} +12a-4b+4 = \\ = \color{blue}{25a^2} \color{red}{-14ab} + \color{green}{2b^2} +12a-4b+4 $$ |
| ⑥ | Combine like terms: $$ \color{blue}{25a^2} \color{red}{-14ab} + \color{green}{2b^2} + \, \color{orange}{ \cancel{12a}} \, \color{red}{-4b} + \color{green}{4} + \color{blue}{9a^2} + \color{red}{6ab} + \color{green}{b^2} \, \color{orange}{ -\cancel{12a}} \, \color{red}{-4b} + \color{green}{4} = \\ = \color{blue}{34a^2} \color{red}{-8ab} + \color{green}{3b^2} \color{red}{-8b} + \color{green}{8} $$ |
| ⑦ | Combine like terms: $$ \color{blue}{34a^2} \, \color{red}{ -\cancel{8ab}} \,+ \color{orange}{3b^2} \color{blue}{-8b} + \color{red}{8} + \color{blue}{16a^2} + \, \color{red}{ \cancel{8ab}} \,+ \color{orange}{b^2} -32a \color{blue}{-8b} + \color{red}{16} = \\ = \color{blue}{50a^2} + \color{orange}{4b^2} -32a \color{blue}{-16b} + \color{red}{24} $$ |
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