$$\dfrac{\dfrac{25}{4}-\dfrac{x}{8}}{\dfrac{x^2}{2}+\dfrac{5}{2}}=\dfrac{-x+50}{4x^2+20}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\frac{25}{4}-\frac{x}{8}}{\frac{x^2}{2}+\frac{5}{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{-x+50}{8}}{\frac{x^2+5}{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-2x+100}{8x^2+40} \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{-x+50}{4x^2+20}\end{aligned} $$ | |
| ① | Subtract $ \dfrac{x}{8} $ from $ \dfrac{25}{4} $ to get $ \dfrac{ \color{purple}{ -x+50 } }{ 8 }$. To subtract raitonal expressions, both fractions must have the same denominator. |
| ② | Add $ \dfrac{x^2}{2} $ and $ \dfrac{5}{2} $ to get $ \dfrac{ x^2 + 5 }{ \color{blue}{ 2 }}$. To add expressions with the same denominators, we add the numerators and write the result over the common denominator. $$ \begin{aligned} \frac{x^2}{2} + \frac{5}{2} & = \frac{x^2}{\color{blue}{2}} + \frac{5}{\color{blue}{2}} =\frac{ x^2 + 5 }{ \color{blue}{ 2 }} \end{aligned} $$ |
| ③ | Divide $ \dfrac{-x+50}{8} $ by $ \dfrac{x^2+5}{2} $ to get $ \dfrac{ -2x+100 }{ 8x^2+40 } $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{-x+50}{8} }{ \frac{\color{blue}{x^2+5}}{\color{blue}{2}} } & \xlongequal{\text{Step 1}} \frac{-x+50}{8} \cdot \frac{\color{blue}{2}}{\color{blue}{x^2+5}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( -x+50 \right) \cdot 2 }{ 8 \cdot \left( x^2+5 \right) } \xlongequal{\text{Step 3}} \frac{ -2x+100 }{ 8x^2+40 } \end{aligned} $$ |