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Question

Answer

$$x^2-7x+\dfrac{10}{x^2}-12x+20=\dfrac{x^4-19x^3+20x^2+10}{x^2}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}x^2-7x+\frac{10}{x^2}-12x+20& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{x^4-7x^3+10}{x^2}-12x+20 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{x^4-19x^3+10}{x^2}+20 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{x^4-19x^3+20x^2+10}{x^2}\end{aligned} $$
①

Add $x^2-7x$ and $ \dfrac{10}{x^2} $ to get $ \dfrac{ \color{purple}{ x^4-7x^3+10 } }{ x^2 }$.

Step 1: Write $ x^2-7x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ x^2 }$.

$$ \begin{aligned} x^2-7x+ \frac{10}{x^2} & \xlongequal{\text{Step 1}} \frac{x^2-7x}{\color{red}{1}} + \frac{10}{x^2} = \frac{ \left( x^2-7x \right) \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} + \frac{ 10 }{ x^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4-7x^3 } }{ x^2 } + \frac{ \color{purple}{ 10 } }{ x^2 }=\frac{ \color{purple}{ x^4-7x^3+10 } }{ x^2 } \end{aligned} $$
②

Subtract $12x$ from $ \dfrac{x^4-7x^3+10}{x^2} $ to get $ \dfrac{ \color{purple}{ x^4-19x^3+10 } }{ x^2 }$.

Step 1: Write $ 12x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^4-7x^3+10}{x^2} -12x & \xlongequal{\text{Step 1}} \frac{x^4-7x^3+10}{x^2} - \frac{12x}{\color{red}{1}} = \frac{ x^4-7x^3+10 }{ x^2 } - \frac{ 12x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4-7x^3+10 } }{ x^2 } - \frac{ \color{purple}{ 12x^3 } }{ x^2 }=\frac{ \color{purple}{ x^4-19x^3+10 } }{ x^2 } \end{aligned} $$
③

Add $ \dfrac{x^4-19x^3+10}{x^2} $ and $ 20 $ to get $ \dfrac{ \color{purple}{ x^4-19x^3+20x^2+10 } }{ x^2 }$.

Step 1: Write $ 20 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^4-19x^3+10}{x^2} +20 & \xlongequal{\text{Step 1}} \frac{x^4-19x^3+10}{x^2} + \frac{20}{\color{red}{1}} = \frac{ x^4-19x^3+10 }{ x^2 } + \frac{ 20 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4-19x^3+10 } }{ x^2 } + \frac{ \color{purple}{ 20x^2 } }{ x^2 }=\frac{ \color{purple}{ x^4-19x^3+20x^2+10 } }{ x^2 } \end{aligned} $$
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