In the quadratic equation `ax^(2)+bx+2=0`, both a and b belong to the set `P={3,4,5}`. How many different quadratic equations can be formed using the values from the set P for a and b so that the roots of the equation are real?

Options

A. 3
B. 5
C. 6
D. 8

Answer

To find how many different quadratic equations can be formed using the values from the set \( P = \{3, 4, 5\} \) for \( a \) and \( b \) such that the roots of the equation \( ax^2 + bx + 2 = 0 \) are real, we need to analyze the condition for real roots.

### Step-by-step Solution:

1. **Identify the Condition for Real Roots**:
The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are real if the discriminant \( D \) is non-negative. The discriminant is given by:
\[ D = b^2 - 4ac \]
For our equation, \( c = 2 \), so the condition becomes:
\[ D = b^2 - 8a \geq 0 \]
This simplifies to:
\[ b^2 \geq 8a \]

2. **Evaluate Possible Values for \( a \) and \( b \)**:
The values of \( a \) and \( b \) can be chosen from the set \( P = \{3, 4, 5\} \). We will evaluate each combination of \( a \) and \( b \).

3. **Check Each Combination**:
- For \( a = 3 \):
- \( b = 3 \): \( 3^2 = 9 \) and \( 8 \times 3 = 24 \) → \( 9 < 24 \) (not valid)
- \( b = 4 \): \( 4^2 = 16 \) and \( 8 \times 3 = 24 \) → \( 16 < 24 \) (not valid)
- \( b = 5 \): \( 5^2 = 25 \) and \( 8 \times 3 = 24 \) → \( 25 \geq 24 \) (valid)

- For \( a = 4 \):
- \( b = 3 \): \( 3^2 = 9 \) and \( 8 \times 4 = 32 \) → \( 9 < 32 \) (not valid)
- \( b = 4 \): \( 4^2 = 16 \) and \( 8 \times 4 = 32 \) → \( 16 < 32 \) (not valid)
- \( b = 5 \): \( 5^2 = 25 \) and \( 8 \times 4 = 32 \) → \( 25 < 32 \) (not valid)

- For \( a = 5 \):
- \( b = 3 \): \( 3^2 = 9 \) and \( 8 \times 5 = 40 \) → \( 9 < 40 \) (not valid)
- \( b = 4 \): \( 4^2 = 16 \) and \( 8 \times 5 = 40 \) → \( 16 < 40 \) (not valid)
- \( b = 5 \): \( 5^2 = 25 \) and \( 8 \times 5 = 40 \) → \( 25 < 40 \) (not valid)

4. **Count the Valid Combinations**:
From our evaluations, the only valid combination is:
- \( (a, b) = (3, 5) \)

Thus, the total number of different quadratic equations that can be formed such that the roots are real is **1**.

### Final Answer:
There is **1** valid quadratic equation that can be formed.