`(3a+9)/(a-3)^2+ (-9)/(3a-9)`
In the expression above, `(a-3)^(2) =6.` What is the value of the expression ?

Answer

To solve the expression \(\frac{3a + 9}{(a - 3)^2} + \frac{-9}{3a - 9}\) given that \((a - 3)^2 = 6\), we will follow these steps:

### Step 1: Understand the given condition
We know that \((a - 3)^2 = 6\). Taking the square root of both sides gives us:
\[ a - 3 = \sqrt{6} \quad \text{or} \quad a - 3 = -\sqrt{6} \]
Thus, we can find two possible values for \(a\):
\[ a = 3 + \sqrt{6} \quad \text{or} \quad a = 3 - \sqrt{6} \]

### Step 2: Substitute \(a\) into the expression
We will evaluate the expression for both values of \(a\).

#### Case 1: \(a = 3 + \sqrt{6}\)

1. Calculate \(3a + 9\):
\[ 3a + 9 = 3(3 + \sqrt{6}) + 9 = 9 + 3\sqrt{6} + 9 = 18 + 3\sqrt{6} \]

2. Substitute into the expression:
\[ \frac{3a + 9}{(a - 3)^2} = \frac{18 + 3\sqrt{6}}{6} = \frac{18}{6} + \frac{3\sqrt{6}}{6} = 3 + \frac{\sqrt{6}}{2} \]

3. Calculate \(3a - 9\):
\[ 3a - 9 = 3(3 + \sqrt{6}) - 9 = 9 + 3\sqrt{6} - 9 = 3\sqrt{6} \]

4. Substitute into the second part of the expression:
\[ \frac{-9}{3a - 9} = \frac{-9}{3\sqrt{6}} = -\frac{3}{\sqrt{6}} = -\frac{3\sqrt{6}}{6} = -\frac{\sqrt{6}}{2} \]

5. Combine both parts:
\[ \left(3 + \frac{\sqrt{6}}{2}\right) + \left(-\frac{\sqrt{6}}{2}\right) = 3 + 0 = 3 \]

#### Case 2: \(a = 3 - \sqrt{6}\)

1. Calculate \(3a + 9\):
\[ 3a + 9 = 3(3 - \sqrt{6}) + 9 = 9 - 3\sqrt{6} + 9 = 18 - 3\sqrt{6} \]

2. Substitute into the expression:
\[ \frac{3a + 9}{(a - 3)^2} = \frac{18 - 3\sqrt{6}}{6} = \frac{18}{6} - \frac{3\sqrt{6}}{6} = 3 - \frac{\sqrt{6}}{2} \]

3. Calculate \(3a - 9\):
\[ 3a - 9 = 3(3 - \sqrt{6}) - 9 = 9 - 3\sqrt{6} - 9 = -3\sqrt{6} \]

4. Substitute into the second part of the expression:
\[ \frac{-9}{3a - 9} = \frac{-9}{-3\sqrt{6}} = \frac{3}{\sqrt{6}} = \frac{3\sqrt{6}}{6} = \frac{\sqrt{6}}{2} \]

5. Combine both parts:
\[ \left(3 - \frac{\sqrt{6}}{2}\right) + \left(\frac{\sqrt{6}}{2}\right) = 3 + 0 = 3 \]

### Final Answer
In both cases, the value of the expression is:
\[ \boxed{3} \]