Question
Answer
$$(-2a+b-0)^2+(b-2)^2+(2a+b-5)^2+(4a+b-7)^2=24a^2+8ab+4b^2-76a-28b+78$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-2a+b-0)^2+(b-2)^2+(2a+b-5)^2+(4a+b-7)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}4a^2-4ab+b^2+b^2-4b+4+4a^2+4ab+b^2-20a-10b+25+16a^2+8ab+b^2-56a-14b+49 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}4a^2-4ab+2b^2-4b+4+4a^2+4ab+b^2-20a-10b+25+16a^2+8ab+b^2-56a-14b+49 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}8a^2+3b^2-20a-14b+29+16a^2+8ab+b^2-56a-14b+49 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}24a^2+8ab+4b^2-76a-28b+78\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{-2a+b0}\right) $ by each term in $ \left( -2a+b0\right) $. $$ \left( \color{blue}{-2a+b0}\right) \cdot \left( -2a+b0\right) = \\ = 4a^2-2ab \cancel{0a}-2ab+b^2 \cancel{0b} \cancel{0a} \cancel{0b}0 $$ |
| ② | Combine like terms: $$ 4a^2 \color{blue}{-2ab} \, \color{red}{ \cancel{0a}} \, \color{blue}{-2ab} +b^2 \, \color{orange}{ \cancel{0b}} \, \, \color{red}{ \cancel{0a}} \, \, \color{orange}{ \cancel{0b}} \,0 = 4a^2 \color{blue}{-4ab} +b^2 $$Find $ \left(b-2\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ b } $ and $ B = \color{red}{ 2 }$. $$ \begin{aligned}\left(b-2\right)^2 = \color{blue}{b^2} -2 \cdot b \cdot 2 + \color{red}{2^2} = b^2-4b+4\end{aligned} $$Multiply each term of $ \left( \color{blue}{2a+b-5}\right) $ by each term in $ \left( 2a+b-5\right) $. $$ \left( \color{blue}{2a+b-5}\right) \cdot \left( 2a+b-5\right) = 4a^2+2ab-10a+2ab+b^2-5b-10a-5b+25 $$ |
| ③ | Combine like terms: $$ 4a^2+ \color{blue}{2ab} \color{red}{-10a} + \color{blue}{2ab} +b^2 \color{green}{-5b} \color{red}{-10a} \color{green}{-5b} +25 = \\ = 4a^2+ \color{blue}{4ab} +b^2 \color{red}{-20a} \color{green}{-10b} +25 $$Multiply each term of $ \left( \color{blue}{4a+b-7}\right) $ by each term in $ \left( 4a+b-7\right) $. $$ \left( \color{blue}{4a+b-7}\right) \cdot \left( 4a+b-7\right) = 16a^2+4ab-28a+4ab+b^2-7b-28a-7b+49 $$ |
| ④ | Combine like terms: $$ 16a^2+ \color{blue}{4ab} \color{red}{-28a} + \color{blue}{4ab} +b^2 \color{green}{-7b} \color{red}{-28a} \color{green}{-7b} +49 = \\ = 16a^2+ \color{blue}{8ab} +b^2 \color{red}{-56a} \color{green}{-14b} +49 $$ |
| ⑤ | Combine like terms: $$ 4a^2-4ab+ \color{blue}{b^2} + \color{blue}{b^2} -4b+4 = 4a^2-4ab+ \color{blue}{2b^2} -4b+4 $$ |
| ⑥ | Combine like terms: $$ \color{blue}{4a^2} \, \color{red}{ -\cancel{4ab}} \,+ \color{orange}{2b^2} \color{blue}{-4b} + \color{red}{4} + \color{blue}{4a^2} + \, \color{red}{ \cancel{4ab}} \,+ \color{orange}{b^2} -20a \color{blue}{-10b} + \color{red}{25} = \\ = \color{blue}{8a^2} + \color{orange}{3b^2} -20a \color{blue}{-14b} + \color{red}{29} $$ |
| ⑦ | Combine like terms: $$ \color{blue}{8a^2} + \color{red}{3b^2} \color{green}{-20a} \color{orange}{-14b} + \color{blue}{29} + \color{blue}{16a^2} +8ab+ \color{red}{b^2} \color{green}{-56a} \color{orange}{-14b} + \color{blue}{49} = \\ = \color{blue}{24a^2} +8ab+ \color{red}{4b^2} \color{green}{-76a} \color{orange}{-28b} + \color{blue}{78} $$ |
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