Let `4x^(2)-4(alpha-2)x + alpha-2=0(alpha in R)` be a quadratic equation. Find the values of `alpha` for which
Both the roots are real and distinct.

Answer

To solve the quadratic equation \( 4x^2 - 4(\alpha - 2)x + (\alpha - 2) = 0 \) for values of \( \alpha \) such that both roots are real and distinct, we need to analyze the discriminant of the equation.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the form \( Ax^2 + Bx + C = 0 \), where:
- \( A = 4 \)
- \( B = -4(\alpha - 2) \)
- \( C = \alpha - 2 \)

2. **Calculate the discriminant**:
The discriminant \( D \) of a quadratic equation is given by:
\[ D = B^2 - 4AC \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ D = \left(-4(\alpha - 2)\right)^2 - 4 \cdot 4 \cdot (\alpha - 2) \]

3. **Simplify the discriminant**:
Calculate \( D \):
\[ D = 16(\alpha - 2)^2 - 16(\alpha - 2) \]
Factor out \( 16 \):
\[ D = 16\left((\alpha - 2)^2 - (\alpha - 2)\right) \]

4. **Set the discriminant greater than zero**:
For the roots to be real and distinct, the discriminant must be greater than zero:
\[ 16\left((\alpha - 2)^2 - (\alpha - 2)\right) > 0 \]
Dividing both sides by 16 (since 16 is positive):
\[ (\alpha - 2)^2 - (\alpha - 2) > 0 \]

5. **Let \( y = \alpha - 2 \)**:
Substitute \( y \) into the inequality:
\[ y^2 - y > 0 \]
Factor the expression:
\[ y(y - 1) > 0 \]

6. **Find the critical points**:
The critical points are \( y = 0 \) and \( y = 1 \). We will analyze the intervals determined by these points: \( (-\infty, 0) \), \( (0, 1) \), and \( (1, \infty) \).

7. **Test the intervals**:
- For \( y < 0 \) (e.g., \( y = -1 \)): \( (-1)(-2) > 0 \) (True)
- For \( 0 < y < 1 \) (e.g., \( y = 0.5 \)): \( (0.5)(-0.5) < 0 \) (False)
- For \( y > 1 \) (e.g., \( y = 2 \)): \( (2)(1) > 0 \) (True)

8. **Conclusion for \( y \)**:
The solution to the inequality \( y(y - 1) > 0 \) is:
\[ y < 0 \quad \text{or} \quad y > 1 \]
Converting back to \( \alpha \):
\[ \alpha - 2 < 0 \quad \Rightarrow \quad \alpha < 2 \]
\[ \alpha - 2 > 1 \quad \Rightarrow \quad \alpha > 3 \]

9. **Final answer**:
Thus, the values of \( \alpha \) for which both roots are real and distinct are:
\[ \alpha \in (-\infty, 2) \cup (3, \infty) \]