The equation `(24x^(2)+25x-47)/(ax-2)=-8x-3-(53)/(ax-2)` is true for all values of `xne(2)/(a)` where a is a constant. What is the value of a?

Options

A. `-16`
B. `-3`
C. 3
D. 16

Answer

To solve the equation

\[ \frac{24x^2 + 25x - 47}{ax - 2} = -8x - 3 - \frac{53}{ax - 2} \]

we will follow these steps:

### Step 1: Combine the Right-Hand Side (RHS)
The RHS has two fractions. To combine them, we need a common denominator, which is \( ax - 2 \).

\[ -8x - 3 - \frac{53}{ax - 2} = \frac{(-8x - 3)(ax - 2) - 53}{ax - 2} \]

### Step 2: Expand the Numerator on the RHS
Now we will expand the numerator:

\[ (-8x - 3)(ax - 2) = -8ax^2 + 16x - 3ax + 6 \]

So, the RHS becomes:

\[ \frac{-8ax^2 + (16 - 3a)x + 6 - 53}{ax - 2} = \frac{-8ax^2 + (16 - 3a)x - 47}{ax - 2} \]

### Step 3: Set the Numerators Equal
Since the denominators are the same, we can set the numerators equal to each other:

\[ 24x^2 + 25x - 47 = -8ax^2 + (16 - 3a)x - 47 \]

### Step 4: Compare Coefficients
Now we will compare the coefficients of \( x^2 \) and \( x \):

1. For \( x^2 \):
\[ 24 = -8a \]

2. For \( x \):
\[ 25 = 16 - 3a \]

### Step 5: Solve for \( a \)
From the first equation:

\[ a = \frac{24}{-8} = -3 \]

Now, let's verify using the second equation:

Substituting \( a = -3 \):

\[ 25 = 16 - 3(-3) \implies 25 = 16 + 9 \implies 25 = 25 \]

Both equations are satisfied.

### Conclusion
The value of \( a \) is

\[ \boxed{-3} \]