Let's solve the given equation step-by-step:
[tex]\[
\frac{4}{x-4} - \frac{7}{x+5} = \frac{66}{x^2 + x - 20}
\][/tex]
### Step 1: Factor the Denominator
Firstly, we should factor the quadratic expression [tex]\( x^2 + x - 20 \)[/tex].
The quadratic [tex]\( x^2 + x - 20 \)[/tex] can be factored as follows:
[tex]\[
x^2 + x - 20 = (x - 4)(x + 5)
\][/tex]
### Step 2: Rewrite the Equation with the Factored Denominator
Rewrite the original equation using the factored form of the quadratic expression:
[tex]\[
\frac{4}{x-4} - \frac{7}{x+5} = \frac{66}{(x-4)(x+5)}
\][/tex]
### Step 3: Find a Common Denominator
The common denominator for the left side of the equation is [tex]\( (x-4)(x+5) \)[/tex]:
[tex]\[
\frac{4(x+5)}{(x-4)(x+5)} - \frac{7(x-4)}{(x-4)(x+5)}
\][/tex]
### Step 4: Combine the Fractions
Combine the fractions on the left side:
[tex]\[
\frac{4(x+5) - 7(x-4)}{(x-4)(x+5)}
\][/tex]
### Step 5: Simplify the Numerator
Distribute and combine like terms in the numerator:
[tex]\[
4(x+5) - 7(x-4) = 4x + 20 - 7x + 28 = -3x + 48
\][/tex]
Thus, the equation now is:
[tex]\[
\frac{-3x + 48}{(x-4)(x+5)} = \frac{66}{(x-4)(x+5)}
\][/tex]
### Step 6: Set the Numerators Equal
Since the denominators are the same, set the numerators equal to each other:
[tex]\[
-3x + 48 = 66
\][/tex]
### Step 7: Solve for [tex]\( x \)[/tex]
Solve the equation for [tex]\( x \)[/tex]:
[tex]\[
-3x + 48 = 66
\][/tex]
Subtract 48 from both sides:
[tex]\[
-3x = 18
\][/tex]
Divide both sides by -3:
[tex]\[
x = -6
\][/tex]
### Step 8: Verify the Solution
Finally, substitute [tex]\( x = -6 \)[/tex] back into the original equation to ensure it does not create any undefined expressions:
- Check that [tex]\( x \neq 4 \)[/tex] and [tex]\( x \neq -5 \)[/tex] (the values that would make the denominators zero):
Since [tex]\( x = -6 \)[/tex] does not make any denominator zero and satisfies the equation,
### Conclusion
Thus, the solution to the equation is:
[tex]\[
x = -6
\][/tex]
