Question
Answer
$$(6+dx)^3-(5+dx)^3=3d^2x^2+33dx+91$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(6+dx)^3-(5+dx)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}216+108dx+18d^2x^2+d^3x^3-(125+75dx+15d^2x^2+d^3x^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}216+108dx+18d^2x^2+d^3x^3-125-75dx-15d^2x^2-d^3x^3 \xlongequal{ } \\[1 em] & \xlongequal{ }216+108dx+18d^2x^2+ \cancel{d^3x^3}-125-75dx-15d^2x^2 -\cancel{d^3x^3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}3d^2x^2+33dx+91\end{aligned} $$ | |
| ① | Find $ \left(6+dx\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 6 $ and $ B = dx $. $$ \left(6+dx\right)^3 = 6^3+3 \cdot 6^2 \cdot dx + 3 \cdot 6 \cdot \left( dx \right)^2+\left( dx \right)^3 = 216+108dx+18d^2x^2+d^3x^3 $$Find $ \left(5+dx\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 5 $ and $ B = dx $. $$ \left(5+dx\right)^3 = 5^3+3 \cdot 5^2 \cdot dx + 3 \cdot 5 \cdot \left( dx \right)^2+\left( dx \right)^3 = 125+75dx+15d^2x^2+d^3x^3 $$ |
| ② | Remove the parentheses by changing the sign of each term within them. $$ - \left( 125+75dx+15d^2x^2+d^3x^3 \right) = -125-75dx-15d^2x^2-d^3x^3 $$ |
| ③ | Combine like terms: $$ \color{blue}{216} + \color{red}{108dx} + \color{green}{18d^2x^2} + \, \color{orange}{ \cancel{d^3x^3}} \, \color{blue}{-125} \color{red}{-75dx} \color{green}{-15d^2x^2} \, \color{orange}{ -\cancel{d^3x^3}} \, = \color{green}{3d^2x^2} + \color{red}{33dx} + \color{blue}{91} $$ |
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