If `a, b in {1, 2, 3, 4}`, then the number of quadratic equations of the form `ax^(2)+bx+1=0`, having real roots is :

Options

A. 6
B. 7
C. 8
D. none of these

Answer

To solve the problem of finding the number of quadratic equations of the form \( ax^2 + bx + 1 = 0 \) that have real roots, we need to use the condition for real roots, which is that the discriminant must be non-negative. The discriminant \( D \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

In our case, \( c = 1 \), so the discriminant becomes:

\[ D = b^2 - 4a \]

For the quadratic equation to have real roots, we need:

\[ b^2 - 4a \geq 0 \]

This can be rearranged to:

\[ b^2 \geq 4a \]

Given that \( a \) and \( b \) can take values from the set \( \{1, 2, 3, 4\} \), we will evaluate the possible values for \( b \) for each value of \( a \).

### Step 1: Evaluate for each value of \( a \)

1. **If \( a = 1 \)**:
\[ b^2 \geq 4 \cdot 1 \implies b^2 \geq 4 \implies b \geq 2 \]
Possible values of \( b \): \( 2, 3, 4 \) (3 values)

2. **If \( a = 2 \)**:
\[ b^2 \geq 4 \cdot 2 \implies b^2 \geq 8 \implies b \geq \sqrt{8} \approx 2.83 \]
Possible values of \( b \): \( 3, 4 \) (2 values)

3. **If \( a = 3 \)**:
\[ b^2 \geq 4 \cdot 3 \implies b^2 \geq 12 \implies b \geq \sqrt{12} \approx 3.46 \]
Possible value of \( b \): \( 4 \) (1 value)

4. **If \( a = 4 \)**:
\[ b^2 \geq 4 \cdot 4 \implies b^2 \geq 16 \implies b \geq 4 \]
Possible value of \( b \): \( 4 \) (1 value)

### Step 2: Count the total number of valid pairs \( (a, b) \)

Now, we will sum the number of valid \( b \) values for each \( a \):

- For \( a = 1 \): 3 values
- For \( a = 2 \): 2 values
- For \( a = 3 \): 1 value
- For \( a = 4 \): 1 value

Total number of valid pairs \( (a, b) \):

\[ 3 + 2 + 1 + 1 = 7 \]

### Conclusion

The total number of quadratic equations of the form \( ax^2 + bx + 1 = 0 \) that have real roots is **7**.

---