If `p ,q , in {1,2,3,4},` then find the number of equations of the form `p x^2+q x+1=0` having real roots.

Options

A. 15
B. 9
C. 8
D. 7

Answer

To find the number of equations of the form \( p x^2 + q x + 1 = 0 \) that have real roots, we need to analyze the discriminant of the quadratic equation. The discriminant \( D \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

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

In our case, \( a = p \), \( b = q \), and \( c = 1 \). Therefore, the discriminant becomes:

\[ D = q^2 - 4p \]

For the quadratic equation to have real roots, the discriminant must be greater than or equal to zero:

\[ q^2 - 4p \geq 0 \]

This can be rearranged to:

\[ q^2 \geq 4p \]

Given that \( p \) and \( q \) can take values from the set \( \{1, 2, 3, 4\} \), we will evaluate each possible pair \( (p, q) \) to see how many satisfy the condition \( q^2 \geq 4p \).

### Step-by-Step Evaluation:

1. **For \( p = 1 \)**:
- \( q = 1 \): \( 1^2 \geq 4 \times 1 \) → \( 1 \geq 4 \) (False)
- \( q = 2 \): \( 2^2 \geq 4 \times 1 \) → \( 4 \geq 4 \) (True)
- \( q = 3 \): \( 3^2 \geq 4 \times 1 \) → \( 9 \geq 4 \) (True)
- \( q = 4 \): \( 4^2 \geq 4 \times 1 \) → \( 16 \geq 4 \) (True)

Valid pairs: \( (1, 2), (1, 3), (1, 4) \)

2. **For \( p = 2 \)**:
- \( q = 1 \): \( 1^2 \geq 4 \times 2 \) → \( 1 \geq 8 \) (False)
- \( q = 2 \): \( 2^2 \geq 4 \times 2 \) → \( 4 \geq 8 \) (False)
- \( q = 3 \): \( 3^2 \geq 4 \times 2 \) → \( 9 \geq 8 \) (True)
- \( q = 4 \): \( 4^2 \geq 4 \times 2 \) → \( 16 \geq 8 \) (True)

Valid pairs: \( (2, 3), (2, 4) \)

3. **For \( p = 3 \)**:
- \( q = 1 \): \( 1^2 \geq 4 \times 3 \) → \( 1 \geq 12 \) (False)
- \( q = 2 \): \( 2^2 \geq 4 \times 3 \) → \( 4 \geq 12 \) (False)
- \( q = 3 \): \( 3^2 \geq 4 \times 3 \) → \( 9 \geq 12 \) (False)
- \( q = 4 \): \( 4^2 \geq 4 \times 3 \) → \( 16 \geq 12 \) (True)

Valid pair: \( (3, 4) \)

4. **For \( p = 4 \)**:
- \( q = 1 \): \( 1^2 \geq 4 \times 4 \) → \( 1 \geq 16 \) (False)
- \( q = 2 \): \( 2^2 \geq 4 \times 4 \) → \( 4 \geq 16 \) (False)
- \( q = 3 \): \( 3^2 \geq 4 \times 4 \) → \( 9 \geq 16 \) (False)
- \( q = 4 \): \( 4^2 \geq 4 \times 4 \) → \( 16 \geq 16 \) (True)

Valid pair: \( (4, 4) \)

### Summary of Valid Pairs:
- From \( p = 1 \): \( (1, 2), (1, 3), (1, 4) \) → 3 pairs
- From \( p = 2 \): \( (2, 3), (2, 4) \) → 2 pairs
- From \( p = 3 \): \( (3, 4) \) → 1 pair
- From \( p = 4 \): \( (4, 4) \) → 1 pair

### Total Valid Pairs:
Total = \( 3 + 2 + 1 + 1 = 7 \)

Thus, the number of equations of the form \( p x^2 + q x + 1 = 0 \) having real roots is **7**.