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 greater than `(1)/(2)`

Answer

To solve the quadratic equation \(4x^2 - 4(\alpha - 2)x + (\alpha - 2) = 0\) for the values of \(\alpha\) such that both roots are greater than \(\frac{1}{2}\), we will follow these steps:

### Step 1: Ensure the roots are real
For the roots of the quadratic equation to be real, the discriminant \(D\) must be non-negative. The discriminant for the given quadratic equation is given by:

\[ D = b^2 - 4ac \]

Here, \(a = 4\), \(b = -4(\alpha - 2)\), and \(c = \alpha - 2\).

Calculating the discriminant:

\[ D = [-4(\alpha - 2)]^2 - 4 \cdot 4 \cdot (\alpha - 2) \]

\[ D = 16(\alpha - 2)^2 - 16(\alpha - 2) \]

Factoring out \(16(\alpha - 2)\):

\[ D = 16(\alpha - 2)[(\alpha - 2) - 1] = 16(\alpha - 2)(\alpha - 3) \]

Setting the discriminant \(D \geq 0\):

\[ 16(\alpha - 2)(\alpha - 3) \geq 0 \]

This gives us two critical points: \(\alpha = 2\) and \(\alpha = 3\). Analyzing the sign of the expression, we find:

- The expression is positive when \(\alpha < 2\) or \(\alpha > 3\).

### Step 2: Ensure both roots are greater than \(\frac{1}{2}\)
For both roots to be greater than \(\frac{1}{2}\), we can use the condition that the value of the quadratic function at \(x = \frac{1}{2}\) must be positive (since the parabola opens upwards).

Calculating \(f\left(\frac{1}{2}\right)\):

\[ f\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^2 - 4(\alpha - 2)\left(\frac{1}{2}\right) + (\alpha - 2) \]

\[ = 4 \cdot \frac{1}{4} - 2(\alpha - 2) + (\alpha - 2) \]

\[ = 1 - 2\alpha + 4 + \alpha - 2 \]

\[ = 3 - \alpha \]

Setting this greater than zero:

\[ 3 - \alpha > 0 \implies \alpha < 3 \]

### Step 3: Combine the conditions
From Step 1, we have \(\alpha < 2\) or \(\alpha > 3\). From Step 2, we have \(\alpha < 3\).

Combining these conditions, we find that the only valid range for \(\alpha\) is:

\[ \alpha < 2 \]

Thus, the values of \(\alpha\) for which both roots of the quadratic equation are greater than \(\frac{1}{2}\) are:

\[ \alpha \in (-\infty, 2) \]