if `(1+k)tan^2x-4tanx-1+k=0` has real roots `tanx_1` and `tanx_2` then

Options

A. `k^(2) le 5`
B. `tan(x_(1) + x_(2)) = 2`
C. for `k = 2, x_(1) = pi//4`
D. all of these

Explanation

To solve the equation \((1+k)\tan^2 x - 4\tan x - (1+k) = 0\) and determine the conditions under which it has real roots, we will follow these steps:

### Step 1: Identify the coefficients of the quadratic equation
The given equation can be rewritten in the standard form \(ax^2 + bx + c = 0\):
- Here, \(a = 1 + k\)
- \(b = -4\)
- \(c = -(1 + k)\)

### Step 2: Use the quadratic formula
The roots of the quadratic equation can be found using the quadratic formula:
\[ \tan x = \frac{-b \pm \sqrt{D}}{2a} \]
where \(D\) is the discriminant given by \(D = b^2 - 4ac\).

### Step 3: Calculate the discriminant
Let's calculate the discriminant \(D\):
\[ D = (-4)^2 - 4(1+k)(-(1+k)) = 16 + 4(1+k)^2 \]
For the quadratic equation to have real roots, the discriminant must be greater than or equal to zero:
\[ 16 + 4(1+k)^2 \geq 0 \]
Since \(16\) and \(4(1+k)^2\) are always non-negative, the discriminant is always greater than or equal to zero for all values of \(k\).

### Step 4: Find the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \(\tan x_1 + \tan x_2 = -\frac{b}{a} = \frac{4}{1+k}\)
- The product of the roots \(\tan x_1 \tan x_2 = \frac{c}{a} = \frac{-(1+k)}{1+k} = -1\)

### Step 5: Find \(\tan(x_1 + x_2)\)
Using the formula for the tangent of a sum:
\[ \tan(x_1 + x_2) = \frac{\tan x_1 + \tan x_2}{1 - \tan x_1 \tan x_2} \]
Substituting the values we found:
\[ \tan(x_1 + x_2) = \frac{\frac{4}{1+k}}{1 - (-1)} = \frac{\frac{4}{1+k}}{2} = \frac{2}{1+k} \]

### Step 6: Determine conditions on \(k\)
To ensure that the quadratic has real roots, we can also analyze the condition on \(k\) derived from the discriminant:
\[ 4(1+k)^2 \geq -16 \]
This condition is always satisfied since \(4(1+k)^2\) is always non-negative.

### Conclusion
The quadratic equation \((1+k)\tan^2 x - 4\tan x - (1+k) = 0\) has real roots for all values of \(k\). The sum of the roots is \(\frac{4}{1+k}\) and the product of the roots is \(-1\).