If `px ^(2)+ qx + r=0` has equal roots then value of r will be `"_______."`

Answer

To solve the problem, we need to find the value of \( r \) in the quadratic equation \( px^2 + qx + r = 0 \) when it has equal roots.

### Step-by-Step Solution:

1. **Identify the Condition for Equal Roots**:
For a quadratic equation \( ax^2 + bx + c = 0 \) to have equal roots, the discriminant must be equal to zero. The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
In our case, \( a = p \), \( b = q \), and \( c = r \).

2. **Set the Discriminant to Zero**:
Since the roots are equal, we set the discriminant to zero:
\[ D = q^2 - 4pr = 0 \]

3. **Rearrange the Equation**:
From the equation \( q^2 - 4pr = 0 \), we can rearrange it to solve for \( r \):
\[ q^2 = 4pr \]

4. **Isolate \( r \)**:
To find \( r \), divide both sides of the equation by \( 4p \):
\[ r = \frac{q^2}{4p} \]

5. **Final Result**:
Thus, the value of \( r \) when the quadratic equation has equal roots is:
\[ r = \frac{q^2}{4p} \]