Find the value of `alpha` for which the equation `(alpha-12)x^(2)+2(alpha-12)x+2=0` has equal roots.

Answer

To find the value of \( \alpha \) for which the equation

\[ (\alpha - 12)x^2 + 2(\alpha - 12)x + 2 = 0 \]

has equal roots, we need to use the condition for equal roots in a quadratic equation. The condition states that the discriminant (denoted as \( \Delta \)) must be equal to zero.

### Step 1: Identify coefficients
In the given quadratic equation, we can identify the coefficients:
- \( A = \alpha - 12 \)
- \( B = 2(\alpha - 12) \)
- \( C = 2 \)

### Step 2: Write the discriminant formula
The discriminant \( \Delta \) for a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:

\[ \Delta = B^2 - 4AC \]

### Step 3: Substitute the coefficients into the discriminant
Substituting the identified coefficients into the discriminant formula:

\[ \Delta = [2(\alpha - 12)]^2 - 4(\alpha - 12)(2) \]

### Step 4: Simplify the discriminant
Now we simplify the expression:

\[ \Delta = 4(\alpha - 12)^2 - 8(\alpha - 12) \]

### Step 5: Factor out common terms
Factoring out \( 4(\alpha - 12) \):

\[ \Delta = 4(\alpha - 12)\left((\alpha - 12) - 2\right) \]

This simplifies to:

\[ \Delta = 4(\alpha - 12)(\alpha - 14) \]

### Step 6: Set the discriminant to zero
For the quadratic to have equal roots, we set the discriminant equal to zero:

\[ 4(\alpha - 12)(\alpha - 14) = 0 \]

### Step 7: Solve for \( \alpha \)
This gives us two factors to solve:

1. \( \alpha - 12 = 0 \) → \( \alpha = 12 \)
2. \( \alpha - 14 = 0 \) → \( \alpha = 14 \)

### Conclusion
The values of \( \alpha \) for which the equation has equal roots are:

\[ \alpha = 12 \quad \text{or} \quad \alpha = 14 \]