Find the values of '`alpha`' for which 6 lies between the roots of the equation `x^2 + 2(a - 3)x + 9 = 0`

Answer

To find the values of `alpha` for which 6 lies between the roots of the equation \( x^2 + 2(a - 3)x + 9 = 0 \), we will follow these steps:

### Step 1: Identify the quadratic equation
The given quadratic equation is:
\[ x^2 + 2(a - 3)x + 9 = 0 \]
Here, the coefficients are:
- \( A = 1 \)
- \( B = 2(a - 3) \)
- \( C = 9 \)

### Step 2: Use the condition for 6 to lie between the roots
For a quadratic equation \( Ax^2 + Bx + C = 0 \), if the roots are \( \alpha \) and \( \beta \), then a number \( k \) lies between the roots if:
\[ f(k) < 0 \]
where \( f(x) = Ax^2 + Bx + C \). In our case, we want to check for \( k = 6 \):
\[ f(6) < 0 \]

### Step 3: Substitute \( x = 6 \) into the quadratic equation
Substituting \( x = 6 \):
\[ f(6) = 1(6^2) + 2(a - 3)(6) + 9 \]
Calculating \( f(6) \):
\[ f(6) = 36 + 12(a - 3) + 9 \]
\[ = 36 + 12a - 36 + 9 \]
\[ = 12a + 9 \]

### Step 4: Set up the inequality
We need:
\[ 12a + 9 < 0 \]

### Step 5: Solve the inequality
Rearranging the inequality:
\[ 12a < -9 \]
\[ a < -\frac{9}{12} \]
\[ a < -\frac{3}{4} \]

### Conclusion
Thus, the values of \( a \) for which 6 lies between the roots of the equation are:
\[ a < -\frac{3}{4} \]