To solve the problem of expressing [tex]\(\frac{2x-2}{x^2 - x - 12}\)[/tex] as the sum of two rational expressions, we can follow these steps:
### Step 1: Factorize the Denominator
The denominator is [tex]\(x^2 - x - 12\)[/tex]. We can factor this quadratic as:
[tex]\[ x^2 - x - 12 = (x - 4)(x + 3) \][/tex]
### Step 2: Write the Given Expression in Terms of Partial Fractions
We want to express:
[tex]\[ \frac{2x - 2}{(x - 4)(x + 3)} \][/tex]
in the form:
[tex]\[ \frac{2x - 2}{(x - 4)(x + 3)} = \frac{A}{x - 4} + \frac{B}{x + 3} \][/tex]
### Step 3: Combine the Partial Fractions
To combine these fractions, we'll get a common denominator:
[tex]\[ \frac{A}{x - 4} + \frac{B}{x + 3} = \frac{A(x + 3) + B(x - 4)}{(x - 4)(x + 3)} \][/tex]
This must equal the original expression:
[tex]\[ \frac{2x - 2}{(x - 4)(x + 3)} \][/tex]
### Step 4: Set Up the Equation
Equating the numerators, we have:
[tex]\[ A(x + 3) + B(x - 4) = 2x - 2 \][/tex]
### Step 5: Expand and Combine Like Terms
Expanding and combining like terms, we get:
[tex]\[ Ax + 3A + Bx - 4B = 2x - 2 \][/tex]
[tex]\[ (A + B)x + (3A - 4B) = 2x - 2 \][/tex]
### Step 6: Match Coefficients
From this, we set up the system of equations by matching coefficients:
1. [tex]\( A + B = 2 \)[/tex] (coefficient of [tex]\(x\)[/tex])
2. [tex]\( 3A - 4B = -2 \)[/tex] (constant term)
### Step 7: Solve the System of Equations
Solve the system of linear equations:
[tex]\[ A + B = 2 \][/tex]
[tex]\[ 3A - 4B = -2 \][/tex]
From our calculations, we find:
[tex]\[ A = \frac{6}{7} \][/tex]
[tex]\[ B = \frac{8}{7} \][/tex]
### Step 8: Substitute [tex]\(A\)[/tex] and [tex]\(B\)[/tex] Back into the Partial Fractions
Substitute the values of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] back into the partial fractions:
[tex]\[ \frac{2x - 2}{(x - 4)(x + 3)} = \frac{\frac{6}{7}}{x - 4} + \frac{\frac{8}{7}}{x + 3} \][/tex]
### Step 9: Final Answer
Rewriting these for clarity, we get:
[tex]\[
\frac{2x - 2}{x^2 - x - 12} = \frac{6}{7(x - 4)} + \frac{8}{7(x + 3)}
\][/tex]
Thus, the two rational expressions are:
[tex]\[ \boxed{\frac{6}{7}} \text{ and } \boxed{\frac{7}{7}} \][/tex]
The final form is:
[tex]\[
\frac{2x - 2}{x^2 - x - 12} = \frac{6}{7(x - 4)} + \frac{8}{7(x + 3)}
\][/tex]
