Find 'a' so that the sum of the roots of the equation `ax^(2)+2x-3a=0` may be equal to their product.

Answer

To find the value of 'a' such that the sum of the roots of the equation \( ax^2 + 2x - 3a = 0 \) is equal to their product, we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \), where:
- \( a = a \)
- \( b = 2 \)
- \( c = -3a \)

### 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 \( S \) is given by \( S = -\frac{b}{a} \)
- The product of the roots \( P \) is given by \( P = \frac{c}{a} \)

### Step 3: Calculate the sum of the roots
Using the formula for the sum of the roots:
\[ S = -\frac{b}{a} = -\frac{2}{a} \]

### Step 4: Calculate the product of the roots
Using the formula for the product of the roots:
\[ P = \frac{c}{a} = \frac{-3a}{a} = -3 \]

### Step 5: Set the sum equal to the product
According to the problem, we need to set the sum equal to the product:
\[ -\frac{2}{a} = -3 \]

### Step 6: Solve for 'a'
To solve for 'a', we can eliminate the negative signs:
\[ \frac{2}{a} = 3 \]
Now, cross-multiply:
\[ 2 = 3a \]
Dividing both sides by 3 gives:
\[ a = \frac{2}{3} \]

### Conclusion
Thus, the value of \( a \) that satisfies the condition is:
\[ \boxed{\frac{2}{3}} \]