The number of integral values of a , `a in (6,100)` for which the equation `[tan x]^(2)+tanx-a=0` has real roots, where [.] denotes greatest integer function is ___________

Answer

To find the number of integral values of \( a \) in the interval \( (6, 100) \) for which the equation \( [\tan x]^2 + \tan x - a = 0 \) has real roots, we can follow these steps:

### Step 1: Understanding the Equation
The equation can be rewritten as:
\[ y^2 + y - a = 0 \]
where \( y = [\tan x] \). For this quadratic equation to have real roots, the discriminant must be non-negative.

### Step 2: Finding the Discriminant
The discriminant \( D \) of the quadratic equation \( y^2 + y - a = 0 \) is given by:
\[ D = b^2 - 4ac = 1^2 - 4(1)(-a) = 1 + 4a \]
For the equation to have real roots, we need:
\[ D \geq 0 \implies 1 + 4a \geq 0 \]
This condition is always satisfied for \( a \geq 0 \).

### Step 3: Finding the Range of \( a \)
Next, we need to ensure that the values of \( y = [\tan x] \) are integers. Since \( [\tan x] \) can take any integer value, we can denote \( y \) as \( n \), where \( n \) is an integer.

### Step 4: Expressing \( a \) in terms of \( n \)
From the equation \( n^2 + n - a = 0 \), we can express \( a \) as:
\[ a = n^2 + n \]
We need to find the values of \( n \) such that \( a \) lies in the interval \( (6, 100) \).

### Step 5: Finding the Limits for \( n \)
We can set up the inequalities:
1. \( n^2 + n > 6 \)
2. \( n^2 + n < 100 \)

#### Solving the first inequality:
\[ n^2 + n - 6 > 0 \]
Factoring gives:
\[ (n - 2)(n + 3) > 0 \]
The critical points are \( n = 2 \) and \( n = -3 \). Testing intervals, we find:
- For \( n < -3 \): positive
- For \( -3 < n < 2 \): negative
- For \( n > 2 \): positive

Thus, \( n > 2 \) or \( n < -3 \).

#### Solving the second inequality:
\[ n^2 + n - 100 < 0 \]
Factoring gives:
\[ (n - 10)(n + 10) < 0 \]
The critical points are \( n = 10 \) and \( n = -10 \). Testing intervals, we find:
- For \( n < -10 \): positive
- For \( -10 < n < 10 \): negative
- For \( n > 10 \): positive

Thus, \( -10 < n < 10 \).

### Step 6: Combining the Results
From the inequalities:
1. \( n > 2 \)
2. \( -10 < n < 10 \)

The combined result gives:
\[ 2 < n < 10 \]
The integer values of \( n \) satisfying this are \( 3, 4, 5, 6, 7, 8, 9 \).

### Step 7: Counting the Values
The integral values of \( n \) are \( 3, 4, 5, 6, 7, 8, 9 \), which gives us a total of:
\[ 7 \text{ values} \]

### Final Answer
The number of integral values of \( a \) in the interval \( (6, 100) \) for which the equation has real roots is \( \boxed{7} \).