If `P = ( x ^(2) - 36)/( x ^(2) - 49) and Q = ( x + 6)/(x - 7) ,` then find the vlaue of `( P )/( Q).`

Options

A. `(x + 6)/(x + 7)`
B. `(x - 6)/( x - 7)`
C. `(x +6)/(x - 7)`
D. `(x - 6)/(x +7)`

Answer

To find the value of \( \frac{P}{Q} \), where \( P = \frac{x^2 - 36}{x^2 - 49} \) and \( Q = \frac{x + 6}{x - 7} \), we will follow these steps:

### Step 1: Simplify \( P \)

The expression for \( P \) is:

\[ P = \frac{x^2 - 36}{x^2 - 49} \]

We can factor both the numerator and the denominator:

- The numerator \( x^2 - 36 \) is a difference of squares, which can be factored as:

\[ x^2 - 36 = (x - 6)(x + 6) \]

- The denominator \( x^2 - 49 \) is also a difference of squares, which can be factored as:

\[ x^2 - 49 = (x - 7)(x + 7) \]

Thus, we can rewrite \( P \):

\[ P = \frac{(x - 6)(x + 6)}{(x - 7)(x + 7)} \]

### Step 2: Write \( Q \)

The expression for \( Q \) is given as:

\[ Q = \frac{x + 6}{x - 7} \]

### Step 3: Find \( \frac{P}{Q} \)

Now we can find \( \frac{P}{Q} \):

\[ \frac{P}{Q} = \frac{\frac{(x - 6)(x + 6)}{(x - 7)(x + 7)}}{\frac{x + 6}{x - 7}} \]

To divide by a fraction, we multiply by its reciprocal:

\[ \frac{P}{Q} = \frac{(x - 6)(x + 6)}{(x - 7)(x + 7)} \cdot \frac{x - 7}{x + 6} \]

### Step 4: Cancel Common Terms

Now we can cancel the common term \( (x + 6) \) from the numerator and denominator:

\[ \frac{P}{Q} = \frac{(x - 6)}{(x + 7)} \]

### Final Result

Thus, the value of \( \frac{P}{Q} \) is:

\[ \frac{P}{Q} = \frac{x - 6}{x + 7} \]