Question
Answer
$$(4w+b-1)^2+(5w+b-3)^2+(6w+b-6)^2+(7w+b-10)^2=4b^2+44bw+126w^2-40b-250w+146$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(4w+b-1)^2+(5w+b-3)^2+(6w+b-6)^2+(7w+b-10)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}b^2+8bw+16w^2-2b-8w+1+b^2+10bw+25w^2-6b-30w+9+b^2+12bw+36w^2-12b-72w+36+b^2+14bw+49w^2-20b-140w+100 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}2b^2+18bw+41w^2-8b-38w+10+b^2+12bw+36w^2-12b-72w+36+b^2+14bw+49w^2-20b-140w+100 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}3b^2+30bw+77w^2-20b-110w+46+b^2+14bw+49w^2-20b-140w+100 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}4b^2+44bw+126w^2-40b-250w+146\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{4w+b-1}\right) $ by each term in $ \left( 4w+b-1\right) $. $$ \left( \color{blue}{4w+b-1}\right) \cdot \left( 4w+b-1\right) = 16w^2+4bw-4w+4bw+b^2-b-4w-b+1 $$ |
| ② | Combine like terms: $$ 16w^2+ \color{blue}{4bw} \color{red}{-4w} + \color{blue}{4bw} +b^2 \color{green}{-b} \color{red}{-4w} \color{green}{-b} +1 = b^2+ \color{blue}{8bw} +16w^2 \color{green}{-2b} \color{red}{-8w} +1 $$Multiply each term of $ \left( \color{blue}{5w+b-3}\right) $ by each term in $ \left( 5w+b-3\right) $. $$ \left( \color{blue}{5w+b-3}\right) \cdot \left( 5w+b-3\right) = 25w^2+5bw-15w+5bw+b^2-3b-15w-3b+9 $$ |
| ③ | Combine like terms: $$ 25w^2+ \color{blue}{5bw} \color{red}{-15w} + \color{blue}{5bw} +b^2 \color{green}{-3b} \color{red}{-15w} \color{green}{-3b} +9 = \\ = b^2+ \color{blue}{10bw} +25w^2 \color{green}{-6b} \color{red}{-30w} +9 $$Multiply each term of $ \left( \color{blue}{6w+b-6}\right) $ by each term in $ \left( 6w+b-6\right) $. $$ \left( \color{blue}{6w+b-6}\right) \cdot \left( 6w+b-6\right) = 36w^2+6bw-36w+6bw+b^2-6b-36w-6b+36 $$ |
| ④ | Combine like terms: $$ 36w^2+ \color{blue}{6bw} \color{red}{-36w} + \color{blue}{6bw} +b^2 \color{green}{-6b} \color{red}{-36w} \color{green}{-6b} +36 = \\ = b^2+ \color{blue}{12bw} +36w^2 \color{green}{-12b} \color{red}{-72w} +36 $$Multiply each term of $ \left( \color{blue}{7w+b-10}\right) $ by each term in $ \left( 7w+b-10\right) $. $$ \left( \color{blue}{7w+b-10}\right) \cdot \left( 7w+b-10\right) = 49w^2+7bw-70w+7bw+b^2-10b-70w-10b+100 $$ |
| ⑤ | Combine like terms: $$ 49w^2+ \color{blue}{7bw} \color{red}{-70w} + \color{blue}{7bw} +b^2 \color{green}{-10b} \color{red}{-70w} \color{green}{-10b} +100 = \\ = b^2+ \color{blue}{14bw} +49w^2 \color{green}{-20b} \color{red}{-140w} +100 $$ |
| ⑥ | Combine like terms: $$ \color{blue}{b^2} + \color{red}{8bw} + \color{green}{16w^2} \color{orange}{-2b} \color{blue}{-8w} + \color{red}{1} + \color{blue}{b^2} + \color{red}{10bw} + \color{green}{25w^2} \color{orange}{-6b} \color{blue}{-30w} + \color{red}{9} = \\ = \color{blue}{2b^2} + \color{red}{18bw} + \color{green}{41w^2} \color{orange}{-8b} \color{blue}{-38w} + \color{red}{10} $$ |
| ⑦ | Combine like terms: $$ \color{blue}{2b^2} + \color{red}{18bw} + \color{green}{41w^2} \color{orange}{-8b} \color{blue}{-38w} + \color{red}{10} + \color{blue}{b^2} + \color{red}{12bw} + \color{green}{36w^2} \color{orange}{-12b} \color{blue}{-72w} + \color{red}{36} = \\ = \color{blue}{3b^2} + \color{red}{30bw} + \color{green}{77w^2} \color{orange}{-20b} \color{blue}{-110w} + \color{red}{46} $$ |
| ⑧ | Combine like terms: $$ \color{blue}{3b^2} + \color{red}{30bw} + \color{green}{77w^2} \color{orange}{-20b} \color{blue}{-110w} + \color{red}{46} + \color{blue}{b^2} + \color{red}{14bw} + \color{green}{49w^2} \color{orange}{-20b} \color{blue}{-140w} + \color{red}{100} = \\ = \color{blue}{4b^2} + \color{red}{44bw} + \color{green}{126w^2} \color{orange}{-40b} \color{blue}{-250w} + \color{red}{146} $$ |
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