Tap the blue circles to see an explanation.
$$ \begin{aligned}x(3 \cdot \frac{(-3x+18)^2}{2}-x\frac{(-3x+18)^2}{4}-\frac{(-3x+18)^3}{18})& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x(3 \cdot \frac{9x^2-108x+324}{2}-x\frac{9x^2-108x+324}{4}-\frac{5832-2916x+486x^2-27x^3}{18}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}x(\frac{27x^2-324x+972}{2}-\frac{9x^3-108x^2+324x}{4}-\frac{5832-2916x+486x^2-27x^3}{18}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}x(\frac{-9x^3+162x^2-972x+1944}{4}-\frac{5832-2916x+486x^2-27x^3}{18}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}x \cdot \frac{-27x^3+486x^2-2916x+5832}{36} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}\frac{-27x^4+486x^3-2916x^2+5832x}{36}\end{aligned} $$ | |
① | Find $ \left(-3x+18\right)^2 $ in two steps. S1: Change all signs inside bracket. S2: Apply formula $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 3x } $ and $ B = \color{red}{ 18 }$. $$ \begin{aligned}\left(-3x+18\right)^2& \xlongequal{ S1 } \left(3x-18\right)^2 \xlongequal{ S2 } \color{blue}{\left( 3x \right)^2} -2 \cdot 3x \cdot 18 + \color{red}{18^2} = \\[1 em] & = 9x^2-108x+324\end{aligned} $$ |
② | Find $ \left(-3x+18\right)^2 $ in two steps. S1: Change all signs inside bracket. S2: Apply formula $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 3x } $ and $ B = \color{red}{ 18 }$. $$ \begin{aligned}\left(-3x+18\right)^2& \xlongequal{ S1 } \left(3x-18\right)^2 \xlongequal{ S2 } \color{blue}{\left( 3x \right)^2} -2 \cdot 3x \cdot 18 + \color{red}{18^2} = \\[1 em] & = 9x^2-108x+324\end{aligned} $$ |
③ | Find $ \left(-3x+18\right)^3 $ in two steps. S1: Swap two terms inside bracket S2: apply formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = 18 $ and $ B = 3x $. $$ \left(-3x+18\right)^3 \xlongequal{ S1 } \left(18-3x\right)^3 = 18^3-3 \cdot 18^2 \cdot 3x + 3 \cdot 18 \cdot \left( 3x \right)^2-\left( 3x \right)^3 = 5832-2916x+486x^2-27x^3 $$ |
④ | Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 3 \cdot \frac{9x^2-108x+324}{2} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{9x^2-108x+324}{2} \xlongequal{\text{Step 2}} \frac{ 3 \cdot \left( 9x^2-108x+324 \right) }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 27x^2-324x+972 }{ 2 } \end{aligned} $$ |
⑤ | Step 1: Write $ x $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} x \cdot \frac{9x^2-108x+324}{4} & \xlongequal{\text{Step 1}} \frac{x}{\color{red}{1}} \cdot \frac{9x^2-108x+324}{4} \xlongequal{\text{Step 2}} \frac{ x \cdot \left( 9x^2-108x+324 \right) }{ 1 \cdot 4 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 9x^3-108x^2+324x }{ 4 } \end{aligned} $$ |
⑥ | Find $ \left(-3x+18\right)^3 $ in two steps. S1: Swap two terms inside bracket S2: apply formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = 18 $ and $ B = 3x $. $$ \left(-3x+18\right)^3 \xlongequal{ S1 } \left(18-3x\right)^3 = 18^3-3 \cdot 18^2 \cdot 3x + 3 \cdot 18 \cdot \left( 3x \right)^2-\left( 3x \right)^3 = 5832-2916x+486x^2-27x^3 $$ |
⑦ | To subtract raitonal expressions, both fractions must have the same denominator. |
⑧ | Find $ \left(-3x+18\right)^3 $ in two steps. S1: Swap two terms inside bracket S2: apply formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = 18 $ and $ B = 3x $. $$ \left(-3x+18\right)^3 \xlongequal{ S1 } \left(18-3x\right)^3 = 18^3-3 \cdot 18^2 \cdot 3x + 3 \cdot 18 \cdot \left( 3x \right)^2-\left( 3x \right)^3 = 5832-2916x+486x^2-27x^3 $$ |
⑨ | To subtract raitonal expressions, both fractions must have the same denominator. |
⑩ | Step 1: Write $ x $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} x \cdot \frac{-27x^3+486x^2-2916x+5832}{36} & \xlongequal{\text{Step 1}} \frac{x}{\color{red}{1}} \cdot \frac{-27x^3+486x^2-2916x+5832}{36} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x \cdot \left( -27x^3+486x^2-2916x+5832 \right) }{ 1 \cdot 36 } \xlongequal{\text{Step 3}} \frac{ -27x^4+486x^3-2916x^2+5832x }{ 36 } \end{aligned} $$ |