Question
Answer
$$(x+4h)^2-4(x+3h)^2+6(x+2h)^2-4(x+h)^2+x^2=0$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+4h)^2-4(x+3h)^2+6(x+2h)^2-4(x+h)^2+x^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x^2+8hx+16h^2-4(x^2+6hx+9h^2)+6(x^2+4hx+4h^2)-4(x^2+2hx+h^2)+x^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^2+8hx+16h^2-(4x^2+24hx+36h^2)+6x^2+24hx+24h^2-(4x^2+8hx+4h^2)+x^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^2+8hx+16h^2-4x^2-24hx-36h^2+6x^2+24hx+24h^2-(4x^2+8hx+4h^2)+x^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-20h^2-16hx-3x^2+6x^2+24hx+24h^2-(4x^2+8hx+4h^2)+x^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}4h^2+8hx+3x^2-(4x^2+8hx+4h^2)+x^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}4h^2+8hx+3x^2-4x^2-8hx-4h^2+x^2 \xlongequal{ } \\[1 em] & \xlongequal{ } \cancel{4h^2}+ \cancel{8hx}+3x^2-4x^2 -\cancel{8hx} -\cancel{4h^2}+x^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}0\end{aligned} $$ | |
| ① | Find $ \left(x+4h\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}{ 4h }$. $$ \begin{aligned}\left(x+4h\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 4h + \color{red}{\left( 4h \right)^2} = x^2+8hx+16h^2\end{aligned} $$Find $ \left(x+3h\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}{ 3h }$. $$ \begin{aligned}\left(x+3h\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 3h + \color{red}{\left( 3h \right)^2} = x^2+6hx+9h^2\end{aligned} $$Find $ \left(x+2h\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}{ 2h }$. $$ \begin{aligned}\left(x+2h\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 2h + \color{red}{\left( 2h \right)^2} = x^2+4hx+4h^2\end{aligned} $$Find $ \left(x+h\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}{ h }$. $$ \begin{aligned}\left(x+h\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot h + \color{red}{h^2} = x^2+2hx+h^2\end{aligned} $$ |
| ② | Multiply $ \color{blue}{4} $ by $ \left( x^2+6hx+9h^2\right) $ $$ \color{blue}{4} \cdot \left( x^2+6hx+9h^2\right) = 4x^2+24hx+36h^2 $$Multiply $ \color{blue}{6} $ by $ \left( x^2+4hx+4h^2\right) $ $$ \color{blue}{6} \cdot \left( x^2+4hx+4h^2\right) = 6x^2+24hx+24h^2 $$Multiply $ \color{blue}{4} $ by $ \left( x^2+2hx+h^2\right) $ $$ \color{blue}{4} \cdot \left( x^2+2hx+h^2\right) = 4x^2+8hx+4h^2 $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 4x^2+24hx+36h^2 \right) = -4x^2-24hx-36h^2 $$ |
| ④ | Combine like terms: $$ \color{blue}{x^2} + \color{red}{8hx} + \color{green}{16h^2} \color{blue}{-4x^2} \color{red}{-24hx} \color{green}{-36h^2} = \color{green}{-20h^2} \color{red}{-16hx} \color{blue}{-3x^2} $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-20h^2} \color{red}{-16hx} \color{green}{-3x^2} + \color{green}{6x^2} + \color{red}{24hx} + \color{blue}{24h^2} = \color{blue}{4h^2} + \color{red}{8hx} + \color{green}{3x^2} $$ |
| ⑥ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 4x^2+8hx+4h^2 \right) = -4x^2-8hx-4h^2 $$ |
| ⑦ | Combine like terms: $$ \, \color{blue}{ \cancel{4h^2}} \,+ \, \color{green}{ \cancel{8hx}} \,+ \color{blue}{3x^2} \color{red}{-4x^2} \, \color{green}{ -\cancel{8hx}} \, \, \color{blue}{ -\cancel{4h^2}} \,+ \color{red}{x^2} = 0 $$ |
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