If `(3x+4)/(x^(2)-3x+2)=A/(x-2)-B/(x-1)` then (A, B) =

Options

A. (7, 10)
B. (10, 7)
C. (10, -7)
D. (-10, 7)

Answer

To solve the equation \(\frac{3x + 4}{x^2 - 3x + 2} = \frac{A}{x - 2} - \frac{B}{x - 1}\), we will follow these steps:

### Step 1: Factor the Denominator
First, we need to factor the denominator \(x^2 - 3x + 2\).
\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]
So, we rewrite the equation as:
\[ \frac{3x + 4}{(x - 1)(x - 2)} = \frac{A}{x - 2} - \frac{B}{x - 1} \]

### Step 2: Combine the Right Side
Next, we will combine the right side over a common denominator:
\[ \frac{A}{x - 2} - \frac{B}{x - 1} = \frac{A(x - 1) - B(x - 2)}{(x - 2)(x - 1)} \]
This gives us:
\[ \frac{3x + 4}{(x - 1)(x - 2)} = \frac{A(x - 1) - B(x - 2)}{(x - 2)(x - 1)} \]

### Step 3: Set the Numerators Equal
Since the denominators are equal, we can set the numerators equal:
\[ 3x + 4 = A(x - 1) - B(x - 2) \]

### Step 4: Expand the Right Side
Now we will expand the right side:
\[ 3x + 4 = Ax - A - Bx + 2B \]
Combining like terms gives:
\[ 3x + 4 = (A - B)x + (2B - A) \]

### Step 5: Compare Coefficients
Now we compare the coefficients of \(x\) and the constant terms:
1. Coefficient of \(x\): \(3 = A - B\) (Equation 1)
2. Constant term: \(4 = 2B - A\) (Equation 2)

### Step 6: Solve the System of Equations
From Equation 1, we can express \(A\) in terms of \(B\):
\[ A = B + 3 \]
Now substitute \(A\) in Equation 2:
\[ 4 = 2B - (B + 3) \]
Simplifying this gives:
\[ 4 = 2B - B - 3 \implies 4 = B - 3 \implies B = 7 \]

### Step 7: Find \(A\)
Now substitute \(B\) back into the equation for \(A\):
\[ A = 7 + 3 = 10 \]

### Final Result
Thus, the values of \(A\) and \(B\) are:
\[ (A, B) = (10, 7) \]