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Question

Answer

$$x^2+\dfrac{25}{x^2}-10x+25=\dfrac{x^4-10x^3+25x^2+25}{x^2}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}x^2+\frac{25}{x^2}-10x+25& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{x^4+25}{x^2}-10x+25 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{x^4-10x^3+25}{x^2}+25 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{x^4-10x^3+25x^2+25}{x^2}\end{aligned} $$
①

Add $x^2$ and $ \dfrac{25}{x^2} $ to get $ \dfrac{ \color{purple}{ x^4+25 } }{ x^2 }$.

Step 1: Write $ x^2 $ 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 $\color{blue}{ x^2 }$.

$$ \begin{aligned} x^2+ \frac{25}{x^2} & \xlongequal{\text{Step 1}} \frac{x^2}{\color{red}{1}} + \frac{25}{x^2} = \frac{ x^2 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} + \frac{ 25 }{ x^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4 } }{ x^2 } + \frac{ \color{purple}{ 25 } }{ x^2 }=\frac{ \color{purple}{ x^4+25 } }{ x^2 } \end{aligned} $$
②

Subtract $10x$ from $ \dfrac{x^4+25}{x^2} $ to get $ \dfrac{ \color{purple}{ x^4-10x^3+25 } }{ x^2 }$.

Step 1: Write $ 10x $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^4+25}{x^2} -10x & \xlongequal{\text{Step 1}} \frac{x^4+25}{x^2} - \frac{10x}{\color{red}{1}} = \frac{ x^4+25 }{ x^2 } - \frac{ 10x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4+25 } }{ x^2 } - \frac{ \color{purple}{ 10x^3 } }{ x^2 }=\frac{ \color{purple}{ x^4-10x^3+25 } }{ x^2 } \end{aligned} $$
③

Add $ \dfrac{x^4-10x^3+25}{x^2} $ and $ 25 $ to get $ \dfrac{ \color{purple}{ x^4-10x^3+25x^2+25 } }{ x^2 }$.

Step 1: Write $ 25 $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^4-10x^3+25}{x^2} +25 & \xlongequal{\text{Step 1}} \frac{x^4-10x^3+25}{x^2} + \frac{25}{\color{red}{1}} = \frac{ x^4-10x^3+25 }{ x^2 } + \frac{ 25 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4-10x^3+25 } }{ x^2 } + \frac{ \color{purple}{ 25x^2 } }{ x^2 }=\frac{ \color{purple}{ x^4-10x^3+25x^2+25 } }{ x^2 } \end{aligned} $$
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