Solve Complex Fraction Equation: (x²-64)/(x-8) = 17(x+8)/(64-x²)

To solve this problem, follow these steps:

• Step 1: Factorize the terms involving squares. Notice that $x^2 - 64 = (x-8)(x+8)$ and $64 - x^2 = (8-x)(8+x)$.
• Step 2: Rewrite both sides of the equation using these factorizations:

$\frac{(x-8)(x+8)}{x-8} = \frac{17(x+8)}{(8-x)(8+x)}$

Upon simplifying, the left side becomes $x+8$ because the $(x-8)$ term cancels out, as long as $x \neq 8$.

$x + 8 = \frac{17(x+8)}{-(x-8)(x+8)}$

• Step 3: Cancel $(x+8)$ from both numerator and denominator on the right side (assuming $x \neq -8$).
• Step 4: This simplifies to:

$x + 8 = \frac{-17}{x-8}$

• Step 5: Multiply both sides by $(x-8)$ to eliminate the denominator:

$(x+8)(x-8) = -17$

$x^2 - 64 = -17$

• Step 6: Rearrange and solve for $x$:

$x^2 = 47$

$x = \pm \sqrt{47}$

Therefore, the solution to the problem is $x = \pm \sqrt{47}$.