Consider the following statements in respect of the quadratic equation `4(x - p) (x - q) - r^(2) = 0` where p.q.r are real numbers
1. The roots are real
2. The roots are equal if p = q and r = 0 Which of the above statements is/are correct ?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

To solve the quadratic equation \( 4(x - p)(x - q) - r^2 = 0 \) and analyze the statements provided, we will follow these steps:

### Step 1: Expand the Quadratic Equation
First, we expand the equation:

\[ 4(x - p)(x - q) - r^2 = 0 \]

Expanding \( 4(x - p)(x - q) \):

\[ = 4(x^2 - (p + q)x + pq) - r^2 \]

This simplifies to:

\[ 4x^2 - 4(p + q)x + 4pq - r^2 = 0 \]

### Step 2: Identify Coefficients
Now, we can identify the coefficients \( a \), \( b \), and \( c \) of the quadratic equation \( ax^2 + bx + c = 0 \):

- \( a = 4 \)
- \( b = -4(p + q) \)
- \( c = 4pq - r^2 \)

### Step 3: Calculate the Discriminant
The discriminant \( D \) of a quadratic equation is given by:

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

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = (-4(p + q))^2 - 4(4)(4pq - r^2) \]

Calculating this gives:

\[ D = 16(p + q)^2 - 16(4pq - r^2) \]

Factoring out 16:

\[ D = 16((p + q)^2 - (4pq - r^2)) \]

### Step 4: Analyze the Discriminant
To determine if the roots are real, we need to check if \( D \geq 0 \):

1. **Roots are Real**:
- For the roots to be real, \( D \geq 0 \).
- Since \( (p + q)^2 \) is always non-negative and \( r^2 \) is also non-negative, \( D \) will be non-negative unless \( (p + q)^2 < 4pq - r^2 \), which can be arranged depending on the values of \( p \), \( q \), and \( r \).

2. **Roots are Equal**:
- The roots are equal if \( D = 0 \).
- This occurs when \( (p + q)^2 = 4pq - r^2 \).
- If \( p = q \) and \( r = 0 \), then \( D = 0 \) holds true.

### Conclusion
- **Statement 1**: The roots are real. This is correct as \( D \geq 0 \) can be satisfied.
- **Statement 2**: The roots are equal if \( p = q \) and \( r = 0 \). This is also correct.

Thus, both statements are correct.

### Final Answer
Both statements are correct, so the answer is **Option C: Both 1 and 2**.

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