Question

In the following exercises, perform the indicated operations.

$\frac{5t^2+4t+3}{t^2-25}-\frac{4t^2-8t-32}{t^2-25}$

Short answer

The expression is simplified as$\frac{t+7}{t-5}$

Step-by-step solution

Step 1. Given information

Expression:

$\frac{5t^2+4t+3}{t^2-25}-\frac{4t^2-8t-32}{t^2-25}$

here denominator in both fraction is same.

Step 2. Subtract the numerator of both fractions by keeping the denominator same.

$$\begin{gathered} \frac{(5t^2+4t+3)-(4t^2-8t-32)}{t^2-25} \\ =\frac{5t^2+4t+3-4t^2+8t+32}{t^2-25} \\ =\frac{t^2+12t+35}{t^2-25} \end{gathered}$$

Step 3. Factorize numerator and denominator 

$$\begin{gathered} \frac{t^2+7t+5t+35}{t^2-5^2} \\ =\frac{t(t+7)+5(t+7)}{(t-5)(t+5)} \\ =\frac{(t+5)(t+7)}{(t-5)(t+5)} \end{gathered}$$

Step 4. Cancel common factor in numerator and denominator 

$\frac{t+7}{t-5}$