Find value of p for which the product of roots of the quadratic equation `px^(2) +6x+ 4p =0` is equal to the sum of the roots.

Answer

To find the value of \( p \) for which the product of the roots of the quadratic equation \( px^2 + 6x + 4p = 0 \) is equal to the sum of the roots, we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = p \)
- \( b = 6 \)
- \( c = 4p \)

### Step 2: Write the formulas for the sum and product of the roots
For a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

### Step 3: Calculate the sum and product of the roots
Using the coefficients from our equation:
- Sum of the roots:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{6}{p} \]
- Product of the roots:
\[ \alpha \beta = \frac{c}{a} = \frac{4p}{p} = 4 \]

### Step 4: Set the product equal to the sum
According to the problem, we need to find \( p \) such that:
\[ \alpha \beta = \alpha + \beta \]
Substituting the expressions we derived:
\[ 4 = -\frac{6}{p} \]

### Step 5: Solve for \( p \)
To solve for \( p \), we can multiply both sides by \( p \) to eliminate the fraction:
\[ 4p = -6 \]
Now, divide both sides by 4:
\[ p = -\frac{6}{4} = -\frac{3}{2} \]

### Conclusion
The value of \( p \) is:
\[ \boxed{-\frac{3}{2}} \quad \text{or} \quad -1.5 \]