what values make the denominators zero 1/(x-2)-4/(x+6) = 8/(x^2+4x-12)

To determine the values of \( x \) that make the denominators zero, we'll analyze each denominator separately.

The equation is:

\[
\frac{1}{x-2} - \frac{4}{x+6} = \frac{8}{x^2 + 4x - 12}
\]

**Step 1: Set each denominator equal to zero.**

1. \( x - 2 = 0 \)
2. \( x + 6 = 0 \)
3. \( x^2 + 4x - 12 = 0 \)

**Step 2: Solve each equation.**

1. \( x - 2 = 0 \)

   \[
   x = 2
   \]

2. \( x + 6 = 0 \)

   \[
   x = -6
   \]

3. \( x^2 + 4x - 12 = 0 \)

   To solve this quadratic equation, we can use the quadratic formula:

   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]

   Here, \( a = 1 \), \( b = 4 \), and \( c = -12 \).

   \[
   x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-12)}}{2(1)}
   \]

   \[
   x = \frac{-4 \pm \sqrt{16 + 48}}{2}
   \]

   \[
   x = \frac{-4 \pm \sqrt{64}}{2}
   \]

   \[
   x = \frac{-4 \pm 8}{2}
   \]

   So, the two solutions are:

   \[
   x = \frac{-4 + 8}{2} = \frac{4}{2} = 2
   \]

   \[
   x = \frac{-4 - 8}{2} = \frac{-12}{2} = -6
   \]

**Conclusion:**

The values of \( x \) that make the denominators zero are \( x = 2 \) and \( x = -6 \).

Do you want more details or have any questions?

Here are 5 related questions:

1. How do you solve for \( x \) when the denominators are not zero?
2. What are the steps to factor the quadratic equation \( x^2 + 4x - 12 \)?
3. How do you determine the domain of a rational function?
4. How can you simplify an equation involving multiple rational expressions?
5. What happens if the numerator of a fraction is also zero at the same points as the denominator?

**Tip:** Always check if a quadratic equation can be factored easily before using the quadratic formula.

Learn how to solve rational equations involving denominators and quadratic expressions. This detailed solution covers finding values of x that make denominators zero and solving quadratic equations. Suitable for high school students.

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