Answer
$$(x+1)^2-(x+7)^2=-12x-48$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+1)^2-(x+7)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x^2+2x+1-(x^2+14x+49) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^2+2x+1-x^2-14x-49 \xlongequal{ } \\[1 em] & \xlongequal{ } \cancel{x^2}+2x+1 -\cancel{x^2}-14x-49 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-12x-48\end{aligned} $$ | |
| ① | Find $ \left(x+1\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$Find $ \left(x+7\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 7 }$. $$ \begin{aligned}\left(x+7\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 7 + \color{red}{7^2} = x^2+14x+49\end{aligned} $$ |
| ② | Remove the parentheses by changing the sign of each term within them. $$ - \left( x^2+14x+49 \right) = -x^2-14x-49 $$ |
| ③ | Combine like terms: $$ \, \color{blue}{ \cancel{x^2}} \,+ \color{green}{2x} + \color{orange}{1} \, \color{blue}{ -\cancel{x^2}} \, \color{green}{-14x} \color{orange}{-49} = \color{green}{-12x} \color{orange}{-48} $$ |
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