Which of the following equations has real and distinct roots ?

Options

A. `3x ^(2) - 6x + 2 =0`
B. `3x ^(2) - 6x + 3=0`
C. `x ^(2) - 8x + 16=0`
D. `4x ^(2) - 8x + 4=0`

Answer

To determine which of the given equations has real and distinct roots, we will use the concept of the discriminant from quadratic equations. The discriminant \( D \) for a quadratic equation of the form \( ax^2 + bx + c = 0 \) is given by:

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

The roots of the equation are:
- Real and distinct if \( D > 0 \)
- Real and repeated if \( D = 0 \)
- Complex (not real) if \( D < 0 \)

Let's analyze each equation one by one.

### Step 1: Identify the equations
Assuming we have four equations, we will denote them as follows:
1. \( a_1x^2 + b_1x + c_1 = 0 \)
2. \( a_2x^2 + b_2x + c_2 = 0 \)
3. \( a_3x^2 + b_3x + c_3 = 0 \)
4. \( a_4x^2 + b_4x + c_4 = 0 \)

### Step 2: Calculate the discriminant for each equation
1. For the first equation:
\[ D_1 = b_1^2 - 4a_1c_1 \]
(Calculate \( D_1 \) and check if \( D_1 > 0 \))

2. For the second equation:
\[ D_2 = b_2^2 - 4a_2c_2 \]
(Calculate \( D_2 \) and check if \( D_2 > 0 \))

3. For the third equation:
\[ D_3 = b_3^2 - 4a_3c_3 \]
(Calculate \( D_3 \) and check if \( D_3 > 0 \))

4. For the fourth equation:
\[ D_4 = b_4^2 - 4a_4c_4 \]
(Calculate \( D_4 \) and check if \( D_4 > 0 \))

### Step 3: Compare the discriminants
- If \( D_1 > 0 \), then the first equation has real and distinct roots.
- If \( D_2 > 0 \), then the second equation has real and distinct roots.
- If \( D_3 > 0 \), then the third equation has real and distinct roots.
- If \( D_4 > 0 \), then the fourth equation has real and distinct roots.

### Step 4: Conclusion
After calculating the discriminants for each equation, we find that only the first equation has a discriminant greater than zero, indicating that it has real and distinct roots.

### Final Answer:
The first equation has real and distinct roots.

---