Question

solve the equation
$\frac{8}{x+4}+1=\frac{5x}{x^{2}-2x-24}$
a $x = 9, -8$
b $x = -9, 8$
c $x = -4, 6$
d no solution

Explanation:

Step1: Factor the denominator

$x^2 - 2x - 24 = (x+4)(x-6)$

Step2: Eliminate denominators

Multiply all terms by $(x+4)(x-6)$:
$8(x-6) + (x+4)(x-6) = 5x$

Step3: Expand all expressions

$8x - 48 + x^2 - 2x - 24 = 5x$

Step4: Simplify to quadratic form

Combine like terms:
$x^2 + 6x - 72 = 5x$
$x^2 + x - 72 = 0$

Step5: Factor the quadratic

$(x+9)(x-8) = 0$

Step6: Check for extraneous solutions

$x=-4$ and $x=6$ make original denominators 0. $x=-9$ and $x=8$ are valid.

Answer:

B. $x = -9, 8$