Find the value of `alpha ` for which roots of equations are equal
`4x ^(2) - 20 x + alpha = 0`

Options

A. 35
B. 40
C. 25
D. 78

Answer

To find the value of \( \alpha \) for which the roots of the equation \( 4x^2 - 20x + \alpha = 0 \) are equal, we need to use the condition for equal roots in a quadratic equation. The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are equal when the discriminant \( D \) is zero.

### Step-by-Step Solution:

1. **Identify the coefficients**:
In the equation \( 4x^2 - 20x + \alpha = 0 \), we have:
- \( a = 4 \)
- \( b = -20 \)
- \( c = \alpha \)

2. **Write the formula for the discriminant**:
The discriminant \( D \) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]

3. **Substitute the values into the discriminant formula**:
Substitute \( a \), \( b \), and \( c \) into the discriminant formula:
\[ D = (-20)^2 - 4 \cdot 4 \cdot \alpha \]

4. **Calculate the discriminant**:
\[ D = 400 - 16\alpha \]

5. **Set the discriminant equal to zero for equal roots**:
For the roots to be equal, we set \( D = 0 \):
\[ 400 - 16\alpha = 0 \]

6. **Solve for \( \alpha \)**:
Rearranging the equation gives:
\[ 16\alpha = 400 \]
Dividing both sides by 16:
\[ \alpha = \frac{400}{16} = 25 \]

### Final Answer:
The value of \( \alpha \) for which the roots of the equation are equal is \( \alpha = 25 \).