Question
Answer
$$4c(cx\cdot(1-x)+1-x)+(c\cdot(1-2x)-1)^2-4=c^2+2c-3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}4c(cx\cdot(1-x)+1-x)+(c\cdot(1-2x)-1)^2-4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}4c(c(x-x^2)+1-x)+(1c-2cx-1)^2-4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}4c(c(x-x^2)+1-x)+4c^2x^2-4c^2x+c^2+4cx-2c+1-4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}4c(1cx-cx^2+1-x)+4c^2x^2-4c^2x+c^2+4cx-2c+1-4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}4c(-cx^2+cx-x+1)+4c^2x^2-4c^2x+c^2+4cx-2c+1-4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-4c^2x^2+4c^2x-4cx+4c+4c^2x^2-4c^2x+c^2+4cx-2c+1-4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}c^2+2c+1-4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}c^2+2c-3\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{x} $ by $ \left( 1-x\right) $ $$ \color{blue}{x} \cdot \left( 1-x\right) = x-x^2 $$Multiply $ \color{blue}{c} $ by $ \left( 1-2x\right) $ $$ \color{blue}{c} \cdot \left( 1-2x\right) = c-2cx $$ |
| ② | Multiply each term of $ \left( \color{blue}{c-2cx-1}\right) $ by each term in $ \left( c-2cx-1\right) $. $$ \left( \color{blue}{c-2cx-1}\right) \cdot \left( c-2cx-1\right) = c^2-2c^2x-c-2c^2x+4c^2x^2+2cx-c+2cx+1 $$ |
| ③ | Combine like terms: $$ c^2 \color{blue}{-2c^2x} \color{red}{-c} \color{blue}{-2c^2x} +4c^2x^2+ \color{green}{2cx} \color{red}{-c} + \color{green}{2cx} +1 = \\ = 4c^2x^2 \color{blue}{-4c^2x} +c^2+ \color{green}{4cx} \color{red}{-2c} +1 $$ |
| ④ | Multiply $ \color{blue}{c} $ by $ \left( x-x^2\right) $ $$ \color{blue}{c} \cdot \left( x-x^2\right) = cx-cx^2 $$ |
| ⑤ | Combine like terms: $$ cx-cx^2+1-x = -cx^2+cx-x+1 $$ |
| ⑥ | Multiply $ \color{blue}{4c} $ by $ \left( -cx^2+cx-x+1\right) $ $$ \color{blue}{4c} \cdot \left( -cx^2+cx-x+1\right) = -4c^2x^2+4c^2x-4cx+4c $$ |
| ⑦ | Combine like terms: $$ \, \color{blue}{ -\cancel{4c^2x^2}} \,+ \, \color{green}{ \cancel{4c^2x}} \, \, \color{blue}{ -\cancel{4cx}} \,+ \color{green}{4c} + \, \color{blue}{ \cancel{4c^2x^2}} \, \, \color{green}{ -\cancel{4c^2x}} \,+c^2+ \, \color{blue}{ \cancel{4cx}} \, \color{green}{-2c} +1 = c^2+ \color{green}{2c} +1 $$ |
| ⑧ | Combine like terms: $$ c^2+2c+ \color{blue}{1} \color{blue}{-4} = c^2+2c \color{blue}{-3} $$ |
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