Let 'k' be a real number. The minimum number of distinct real roots possible of the equation `(3x^2+kx+3)(x^2+kx-1)=0` is

Options

A. 0
B. 2
C. 3
D. 4

Answer

To find the minimum number of distinct real roots of the equation \((3x^2 + kx + 3)(x^2 + kx - 1) = 0\), we will analyze the two quadratic equations separately.

### Step 1: Identify the two quadratic equations
The given equation can be split into two separate equations:
1. \(3x^2 + kx + 3 = 0\)
2. \(x^2 + kx - 1 = 0\)

### Step 2: Analyze the first quadratic equation
For the first equation \(3x^2 + kx + 3 = 0\), we need to determine the conditions for it to have distinct real roots. The condition for a quadratic equation \(ax^2 + bx + c = 0\) to have distinct real roots is that the discriminant \(D\) must be greater than 0:
\[ D = b^2 - 4ac > 0 \]
Here, \(a = 3\), \(b = k\), and \(c = 3\). Thus, the discriminant becomes:
\[ D_1 = k^2 - 4(3)(3) = k^2 - 36 \]
Setting the discriminant greater than zero gives:
\[ k^2 - 36 > 0 \]
This simplifies to:
\[ k^2 > 36 \]
Taking the square root of both sides, we find:
\[ |k| > 6 \]
This means \(k\) must be in the intervals:
\[ k < -6 \quad \text{or} \quad k > 6 \]

### Step 3: Analyze the second quadratic equation
Now, we analyze the second equation \(x^2 + kx - 1 = 0\). The discriminant for this equation is:
\[ D_2 = k^2 - 4(1)(-1) = k^2 + 4 \]
Since \(k^2 + 4\) is always greater than 0 for all real \(k\), this equation will always have two distinct real roots.

### Step 4: Combine the results
Now we combine the results from both equations:
- The first equation \(3x^2 + kx + 3 = 0\) has:
- Two distinct real roots if \(|k| > 6\).
- No real roots if \(-6 < k < 6\) (the roots are equal).

- The second equation \(x^2 + kx - 1 = 0\) always has two distinct real roots for any real \(k\).

### Step 5: Determine the minimum number of distinct real roots
1. If \(k < -6\) or \(k > 6\), both equations have 2 distinct real roots each, giving a total of 4 distinct real roots.
2. If \(-6 < k < 6\), the first equation has no real roots (0 roots), while the second equation has 2 distinct real roots.

Thus, the minimum number of distinct real roots possible for the entire equation \((3x^2 + kx + 3)(x^2 + kx - 1) = 0\) is:
\[ \text{Minimum number of distinct real roots} = 2 \]

### Final Answer
The minimum number of distinct real roots possible of the equation is **2**.