If the roots of the quadratic equation `x^2 - ax + 2b = 0` are prime numbers, then the value of (a-b) is

Answer

To solve the problem, we start with the quadratic equation given:

\[ x^2 - ax + 2b = 0 \]

We know that the roots of this equation are prime numbers. Let's denote the roots by \( \alpha \) and \( \beta \).

### Step 1: Use Vieta's Formulas
According to Vieta's formulas, for a quadratic equation of the form \( x^2 + px + q = 0 \):
- The sum of the roots \( \alpha + \beta = a \)
- The product of the roots \( \alpha \beta = 2b \)

### Step 2: Assume the Roots are Prime Numbers
Since \( \alpha \) and \( \beta \) are prime numbers, we can consider the simplest case where one of the roots is the smallest prime number, which is 2.

Let’s assume:
- \( \alpha = 2 \)
- \( \beta = p \) (where \( p \) is another prime number)

### Step 3: Calculate the Product of the Roots
From Vieta's formulas, we have:
\[ \alpha \beta = 2b \]
Substituting \( \alpha = 2 \) and \( \beta = p \):
\[ 2p = 2b \]

Dividing both sides by 2 gives:
\[ p = b \]

### Step 4: Calculate the Sum of the Roots
Now, using the sum of the roots:
\[ \alpha + \beta = a \]
Substituting \( \alpha = 2 \) and \( \beta = p \):
\[ 2 + p = a \]

### Step 5: Substitute \( b \) into the Equation
Since we have \( p = b \), we can substitute \( b \) into the equation for \( a \):
\[ a = 2 + b \]

### Step 6: Calculate \( a - b \)
Now, we need to find \( a - b \):
\[ a - b = (2 + b) - b \]
This simplifies to:
\[ a - b = 2 \]

### Final Answer
Thus, the value of \( a - b \) is:
\[ \boxed{2} \]

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