Question

solve the equation by the method of your choice.\\(\frac{3}{x - 5}+\frac{8}{x - 4}=\frac{x^{2}-34}{x^{2}-9x + 20}\\)\\(\\)\\(\\)the solution set is \\(\boxed{}\\).\\(\\)\\(\\) (simplify your answer. use a comma to separate answers as needed.)

Explanation:

Step1: Factor the right denominator

Notice that $x^2 - 9x + 20 = (x-5)(x-4)$

Step2: Multiply by common denominator

Multiply all terms by $(x-5)(x-4)$ to eliminate fractions:
$3(x-4) + 8(x-5) = x^2 - 34$

Step3: Expand left-hand side

$3x - 12 + 8x - 40 = x^2 - 34$

Step4: Combine like terms

$11x - 52 = x^2 - 34$

Step5: Rearrange to quadratic form

$x^2 - 11x + 18 = 0$

Step6: Factor the quadratic

$(x-2)(x-9) = 0$

Step7: Solve for x and check restrictions

Set each factor to 0: $x=2$ or $x=9$. Check that these do not make original denominators 0 (they don't, since $2
eq4,5$ and $9
eq4,5$).

Answer:

2, 9