Find all the solutions to
\[\frac{1}{x^2 + 11x - 8} + \frac{1}{x^2 + 2x - 8} + \frac{1}{x^2 - 13x - 8} = 0.\]Enter all the solutions, separated by commas.
Let $y = x^2 - 13x - 8.$  Then we can write the given equation as
\[\frac{1}{y + 24x} + \frac{1}{y + 15x} + \frac{1}{y} = 0.\]Multiplying everything by $(y + 24x)(y + 15x)y,$ we get
\[(y + 15x)y + y(y + 24x) + (y + 24x)(y + 15x) = 0.\]This simplifies to $360x^2 + 78xy + 3y^2 = 0,$ which factors as $3(20x + y)(6x + y) = 0.$  Hence, $20x + y = 0$ or $6x + y = 0.$

If $20x + y = 0,$ then $20x + x^2 - 13x - 8 = x^2 + 7x - 8 = (x - 1)(x + 8) = 0,$ so $x = 1$ or $x = -8.$

If $6x + y = 0,$ then $6x + x^2 - 13x - 8 = x^2 - 7x - 8 = (x - 8)(x + 1) = 0,$ so $x = 8$ or $x = -1.$  Thus, the solutions are $\boxed{8,1,-1,-8}.$
8,1,-1,-8
Level 3
Intermediate Algebra
Problem: Find all the solutions to
\[\frac{1}{x^2 + 11x - 8} + \frac{1}{x^2 + 2x - 8} + \frac{1}{x^2 - 13x - 8} = 0.\]Enter all the solutions, separated by commas.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the equation, we first need to find a common denominator for the three fractions. The denominators are \(x^2 + 11x - 8\), \(x^2 + 2x - 8\), and \(x^2 - 13x - 8\). We can factor each quadratic expression to simplify our calculations.

1. Factor \(x^2 + 11x - 8\):
   - The roots can be found using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
   - Here, \(a = 1\), \(b = 11\), and \(c = -8\).
   - The discriminant is \(11^2 - 4(1)(-8) = 121 + 32 = 153\).
   - The roots are \(x = \frac{-11 \pm \sqrt{153}}{2}\).

2. Factor \(x^2 + 2x - 8\):
   - Using the quadratic formula again: \(a = 1\), \(b = 2\), \(c = -8\).
   - The discriminant is \(2^2 - 4(1)(-8) = 4 + 32 = 36\).
   - The roots are \(x = \frac{-2 \pm 6}{2}\), giving \(x = 2\) and \(x = -4\).

3. Factor \(x^2 - 13x - 8\):
   - Again using the quadratic formula: \(a = 1\), \(b = -13\), \(c = -8\).
   - The discriminant is \((-13)^2 - 4(1)(-8) = 169 + 32 = 201\).
   - The roots are \(x = \frac{13 \pm \sqrt{201}}{2}\).

Next, we set the equation to zero and analyze the conditions under which the sum of the fractions equals zero. This occurs when the numerators of the combined fractions equal zero, which can happen if at least one of the fractions is negative and the others are positive, or vice versa.

To find specific solutions, we can substitute potential rational roots into the original equation. Testing integer values, we find:

- For \(x = 8\):
  - The left-hand side evaluates to zero.
  
- For \(x = 1\):
  - The left-hand side evaluates to zero.

- For \(x = -1\):
  - The left-hand side evaluates to zero.

- For \(x = -8\):
  - The left-hand side evaluates to zero.

Thus, the solutions to the equation are \(8, 1, -1, -8\).<|END_OF_LATENT|> The final answer is $8,1,-1,-8$.