Question
\( \begin{array}{rlr}-\frac{c}{a}+\frac{b^{2}}{4 a^{2}} & =\quad \text { (make both fractions have the same denominator) } \\ & =\quad \text { (combine to form a single fraction) } \\ & \end{array} \)
Ask by Frank Sullivan. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
-\frac{c}{a} + \frac{b^2}{4a^2} = \frac{-4ac + b^2}{4a^2}
\]
Solution
To combine the fractions \(-\frac{c}{a} + \frac{b^2}{4a^2}\) into a single fraction, we will follow these steps:
1. **Identify the denominators**: The first fraction has a denominator of \(a\) and the second fraction has a denominator of \(4a^2\).
2. **Find a common denominator**: The least common denominator (LCD) for \(a\) and \(4a^2\) is \(4a^2\).
3. **Rewrite each fraction with the common denominator**:
- For the first fraction \(-\frac{c}{a}\), we need to multiply the numerator and denominator by \(4a\) to get the common denominator:
\[
-\frac{c}{a} = -\frac{c \cdot 4a}{a \cdot 4a} = -\frac{4ac}{4a^2}
\]
- The second fraction \(\frac{b^2}{4a^2}\) already has the common denominator.
4. **Combine the fractions**:
Now we can combine the two fractions:
\[
-\frac{4ac}{4a^2} + \frac{b^2}{4a^2} = \frac{-4ac + b^2}{4a^2}
\]
Now, we have successfully combined the fractions into a single fraction. The final result is:
\[
-\frac{c}{a} + \frac{b^2}{4a^2} = \frac{-4ac + b^2}{4a^2}
\]
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution Bonus Knowledge
To make both fractions have the same denominator, you’ll want to rewrite \(-\frac{c}{a}\) so that it has the denominator \(4a^2\). Multiply the numerator and denominator by \(4a\): \(-\frac{c}{a} = -\frac{c \cdot 4a}{a \cdot 4a} = -\frac{4ac}{4a^2}\). Now both fractions, \(-\frac{4ac}{4a^2}\) and \(\frac{b^{2}}{4a^{2}}\), have the same denominator. Combining these fractions into a single fraction gives you: \(-\frac{4ac}{4a^2} + \frac{b^{2}}{4a^{2}} = \frac{b^{2} - 4ac}{4a^2}\). So, the final form is: \(\frac{b^{2} - 4ac}{4a^2}\).
