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 positive

Answer

To find the values of \( \alpha \) for which both roots of the quadratic equation \( 4x^2 - 4(\alpha - 2)x + \alpha - 2 = 0 \) are positive, we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation can be expressed in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 4 \)
- \( b = -4(\alpha - 2) = -4\alpha + 8 \)
- \( c = \alpha - 2 \)

### Step 2: Ensure the discriminant is non-negative
For the quadratic equation to have real roots, the discriminant \( D \) must be greater than or equal to zero. The discriminant is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-4\alpha + 8)^2 - 4(4)(\alpha - 2) \]
Calculating \( D \):
\[ D = (16\alpha^2 - 64\alpha + 64) - 16(\alpha - 2) \]
\[ D = 16\alpha^2 - 64\alpha + 64 - 16\alpha + 32 \]
\[ D = 16\alpha^2 - 80\alpha + 96 \]
Setting the discriminant \( D \geq 0 \):
\[ 16\alpha^2 - 80\alpha + 96 \geq 0 \]
Dividing the entire inequality by 16:
\[ \alpha^2 - 5\alpha + 6 \geq 0 \]

### Step 3: Factor the quadratic
Factoring the quadratic:
\[ (\alpha - 2)(\alpha - 3) \geq 0 \]
The roots of the equation are \( \alpha = 2 \) and \( \alpha = 3 \).

### Step 4: Analyze the sign of the quadratic
Using a number line to test the intervals:
- For \( \alpha < 2 \): Both factors are negative, so the product is positive.
- For \( 2 < \alpha < 3 \): One factor is positive, and the other is negative, so the product is negative.
- For \( \alpha > 3 \): Both factors are positive, so the product is positive.

Thus, the solution to the inequality is:
\[ \alpha \leq 2 \quad \text{or} \quad \alpha \geq 3 \]

### Step 5: Ensure both roots are positive
For both roots to be positive, we also need to check the condition:
\[ -\frac{b}{2a} > 0 \]
Calculating:
\[ -\frac{-4(\alpha - 2)}{2 \cdot 4} > 0 \]
\[ \frac{4(\alpha - 2)}{8} > 0 \]
\[ \alpha - 2 > 0 \implies \alpha > 2 \]

### Step 6: Combine conditions
From the conditions:
1. \( \alpha \leq 2 \) or \( \alpha \geq 3 \)
2. \( \alpha > 2 \)

The valid range for \( \alpha \) is:
\[ \alpha \geq 3 \]

### Final Answer
The values of \( \alpha \) for which both roots are positive are:
\[ \alpha \in [3, \infty) \]