If the quadratic equation `px^(2)- 2 sqrt5px+15`= 0, has two equal roots then find the value of p.

Answer

To solve the quadratic equation \( px^2 - 2\sqrt{5}px + 15 = 0 \) for the value of \( p \) when it has two equal roots, we will use the condition that the discriminant of the quadratic equation must be zero.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). In our case:
- \( a = p \)
- \( b = -2\sqrt{5}p \)
- \( c = 15 \)

2. **Write the discriminant formula**:
The discriminant \( D \) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]
For the roots to be equal, we need:
\[ D = 0 \]

3. **Substitute the values into the discriminant**:
Substitute \( a \), \( b \), and \( c \) into the discriminant formula:
\[ D = (-2\sqrt{5}p)^2 - 4(p)(15) \]

4. **Simplify the expression**:
Calculate \( (-2\sqrt{5}p)^2 \):
\[ D = 4 \cdot 5 \cdot p^2 - 60p \]
This simplifies to:
\[ D = 20p^2 - 60p \]

5. **Set the discriminant to zero**:
To find \( p \), set the discriminant equal to zero:
\[ 20p^2 - 60p = 0 \]

6. **Factor the equation**:
Factor out the common term:
\[ 20p(p - 3) = 0 \]

7. **Solve for \( p \)**:
Set each factor to zero:
- \( 20p = 0 \) gives \( p = 0 \)
- \( p - 3 = 0 \) gives \( p = 3 \)

8. **Conclusion**:
Since \( p = 0 \) would make the equation not quadratic, we discard it. Therefore, the value of \( p \) is:
\[ p = 3 \]

### Final Answer:
The value of \( p \) is \( 3 \).