The value of 'a' for which the equation `ax^(2)-2sqrt(5)x+4=0` has equal roots ,is

Options

A. `(5)/(4)`
B. `(4)/(5)`
C. `-(5)/(4)`
D. `-(5)/(3)`

Answer

To find the value of 'a' for which the equation \( ax^2 - 2\sqrt{5}x + 4 = 0 \) has equal roots, we will use the condition that the discriminant of a quadratic equation must be zero for the roots to be equal.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = a \)
- \( b = -2\sqrt{5} \)
- \( c = 4 \)

2. **Write the discriminant condition**:
The condition for equal roots is given by the discriminant \( D \) being equal to zero:
\[ D = b^2 - 4ac = 0 \]

3. **Substitute the values of b and c into the discriminant**:
Substitute \( b = -2\sqrt{5} \) and \( c = 4 \) into the discriminant formula:
\[ D = (-2\sqrt{5})^2 - 4(a)(4) \]

4. **Calculate \( b^2 \)**:
Calculate \( (-2\sqrt{5})^2 \):
\[ (-2\sqrt{5})^2 = 4 \cdot 5 = 20 \]

5. **Set up the equation**:
Substitute this back into the discriminant equation:
\[ 20 - 16a = 0 \]

6. **Solve for 'a'**:
Rearranging the equation gives:
\[ 20 = 16a \]
Dividing both sides by 16:
\[ a = \frac{20}{16} = \frac{5}{4} \]

### Final Answer:
The value of \( a \) for which the equation has equal roots is \( \frac{5}{4} \).

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