Question
Answer
$$(2w+b-5)^2+(10w+b-9)^2+(15w+b-12)^2=3b^2+54bw+329w^2-52b-560w+250$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(2w+b-5)^2+(10w+b-9)^2+(15w+b-12)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}b^2+4bw+4w^2-10b-20w+25+b^2+20bw+100w^2-18b-180w+81+b^2+30bw+225w^2-24b-360w+144 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}2b^2+24bw+104w^2-28b-200w+106+b^2+30bw+225w^2-24b-360w+144 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}3b^2+54bw+329w^2-52b-560w+250\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{2w+b-5}\right) $ by each term in $ \left( 2w+b-5\right) $. $$ \left( \color{blue}{2w+b-5}\right) \cdot \left( 2w+b-5\right) = 4w^2+2bw-10w+2bw+b^2-5b-10w-5b+25 $$ |
| ② | Combine like terms: $$ 4w^2+ \color{blue}{2bw} \color{red}{-10w} + \color{blue}{2bw} +b^2 \color{green}{-5b} \color{red}{-10w} \color{green}{-5b} +25 = \\ = b^2+ \color{blue}{4bw} +4w^2 \color{green}{-10b} \color{red}{-20w} +25 $$Multiply each term of $ \left( \color{blue}{10w+b-9}\right) $ by each term in $ \left( 10w+b-9\right) $. $$ \left( \color{blue}{10w+b-9}\right) \cdot \left( 10w+b-9\right) = 100w^2+10bw-90w+10bw+b^2-9b-90w-9b+81 $$ |
| ③ | Combine like terms: $$ 100w^2+ \color{blue}{10bw} \color{red}{-90w} + \color{blue}{10bw} +b^2 \color{green}{-9b} \color{red}{-90w} \color{green}{-9b} +81 = \\ = b^2+ \color{blue}{20bw} +100w^2 \color{green}{-18b} \color{red}{-180w} +81 $$Multiply each term of $ \left( \color{blue}{15w+b-12}\right) $ by each term in $ \left( 15w+b-12\right) $. $$ \left( \color{blue}{15w+b-12}\right) \cdot \left( 15w+b-12\right) = 225w^2+15bw-180w+15bw+b^2-12b-180w-12b+144 $$ |
| ④ | Combine like terms: $$ 225w^2+ \color{blue}{15bw} \color{red}{-180w} + \color{blue}{15bw} +b^2 \color{green}{-12b} \color{red}{-180w} \color{green}{-12b} +144 = \\ = b^2+ \color{blue}{30bw} +225w^2 \color{green}{-24b} \color{red}{-360w} +144 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{b^2} + \color{red}{4bw} + \color{green}{4w^2} \color{orange}{-10b} \color{blue}{-20w} + \color{red}{25} + \color{blue}{b^2} + \color{red}{20bw} + \color{green}{100w^2} \color{orange}{-18b} \color{blue}{-180w} + \color{red}{81} = \\ = \color{blue}{2b^2} + \color{red}{24bw} + \color{green}{104w^2} \color{orange}{-28b} \color{blue}{-200w} + \color{red}{106} $$ |
| ⑥ | Combine like terms: $$ \color{blue}{2b^2} + \color{red}{24bw} + \color{green}{104w^2} \color{orange}{-28b} \color{blue}{-200w} + \color{red}{106} + \color{blue}{b^2} + \color{red}{30bw} + \color{green}{225w^2} \color{orange}{-24b} \color{blue}{-360w} + \color{red}{144} = \\ = \color{blue}{3b^2} + \color{red}{54bw} + \color{green}{329w^2} \color{orange}{-52b} \color{blue}{-560w} + \color{red}{250} $$ |
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