To solve the equation
\[ \frac{9}{x} + \frac{9}{x - 2} = 12, \]
we can start by finding a common denominator for the left-hand side. The common denominator will be \(x(x - 2)\). Rewriting the equation with the common denominator, we have:
\[ \frac{9(x - 2) + 9x}{x(x - 2)} = 12. \]
Now simplifying the numerator:
\[ 9(x - 2) + 9x = 9x - 18 + 9x = 18x - 18. \]
So the equation becomes:
\[ \frac{18x - 18}{x(x - 2)} = 12. \]
Next, we can eliminate the fraction by multiplying both sides by \(x(x - 2)\):
\[ 18x - 18 = 12x(x - 2). \]
Expanding the right-hand side gives:
\[ 18x - 18 = 12x^2 - 24x. \]
Now, we can rearrange all terms to one side of the equation:
\[ 0 = 12x^2 - 24x - 18x + 18, \] which simplifies to:
\[ 0 = 12x^2 - 42x + 18. \]
Dividing the entire equation by 6 to simplify yields:
\[ 0 = 2x^2 - 7x + 3. \]
Next, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \(a = 2\), \(b = -7\), and \(c = 3\).
Calculating the discriminant:
\[ b^2 - 4ac = (-7)^2 - 4 \cdot 2 \cdot 3 = 49 - 24 = 25. \]
Now we can find the solutions:
\[ x = \frac{-(-7) \pm \sqrt{25}}{2 \cdot 2} = \frac{7 \pm 5}{4}. \]
Calculating the two possible values for \(x\):
- \( x = \frac{7 + 5}{4} = \frac{12}{4} = 3 \)
- \( x = \frac{7 - 5}{4} = \frac{2}{4} = \frac{1}{2} \)
Thus, the solutions to the equation are:
\[ x = 3 \quad \text{and} \quad x = \frac{1}{2}. \]
Finally, we should check if either \(x\) values make the original equation undefined (division by zero):
- For \(x = 3\), \( x - 2 \) is \(1\) (valid).
- For \(x = \frac{1}{2}\), \( x - 2 \) is \(-\frac{3}{2}\) (valid).
Both solutions are valid, thus:
The solutions are \(x = 3\) and \(x = \frac{1}{2}\).