If `alpha,beta(alpha lt beta)` are the real roots of equation `x^2-(k+4)x+k^2-12=0` such theta `4 in (alpha,beta)`, then the number of integral value of `k` is equal to

Options

A. 4
B. 5
C. 6
D. 7

Answer

To solve the problem, we need to find the number of integral values of \( k \) such that the quadratic equation \( x^2 - (k+4)x + (k^2 - 12) = 0 \) has real roots \( \alpha \) and \( \beta \) with \( \alpha < 4 < \beta \).

### Step 1: Identify the quadratic equation
The given quadratic equation is:
\[ x^2 - (k+4)x + (k^2 - 12) = 0 \]

### Step 2: Calculate the discriminant
For the roots to be real, the discriminant \( D \) must be non-negative:
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = -(k+4) \), and \( c = k^2 - 12 \). Thus,
\[ D = (k+4)^2 - 4 \cdot 1 \cdot (k^2 - 12) \]
\[ D = (k+4)^2 - 4(k^2 - 12) \]
\[ D = k^2 + 8k + 16 - 4k^2 + 48 \]
\[ D = -3k^2 + 8k + 64 \]
Setting \( D \geq 0 \):
\[ -3k^2 + 8k + 64 \geq 0 \]

### Step 3: Solve the quadratic inequality
To find the values of \( k \), we first find the roots of the equation:
\[ -3k^2 + 8k + 64 = 0 \]
Using the quadratic formula \( k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ k = \frac{-8 \pm \sqrt{8^2 - 4 \cdot (-3) \cdot 64}}{2 \cdot (-3)} \]
\[ k = \frac{-8 \pm \sqrt{64 + 768}}{-6} \]
\[ k = \frac{-8 \pm \sqrt{832}}{-6} \]
\[ k = \frac{-8 \pm 4\sqrt{52}}{-6} \]
\[ k = \frac{-8 \pm 4 \cdot 2\sqrt{13}}{-6} \]
\[ k = \frac{4 \pm 2\sqrt{13}}{3} \]
Let \( k_1 = \frac{4 - 2\sqrt{13}}{3} \) and \( k_2 = \frac{4 + 2\sqrt{13}}{3} \).

### Step 4: Determine the interval for \( k \)
Since the parabola opens downwards (the coefficient of \( k^2 \) is negative), the inequality \( -3k^2 + 8k + 64 \geq 0 \) holds between the roots:
\[ \frac{4 - 2\sqrt{13}}{3} \leq k \leq \frac{4 + 2\sqrt{13}}{3} \]

### Step 5: Find the approximate values of the roots
Calculating \( \sqrt{13} \approx 3.605 \):
\[ k_1 \approx \frac{4 - 2 \cdot 3.605}{3} \approx \frac{4 - 7.21}{3} \approx \frac{-3.21}{3} \approx -1.07 \]
\[ k_2 \approx \frac{4 + 2 \cdot 3.605}{3} \approx \frac{4 + 7.21}{3} \approx \frac{11.21}{3} \approx 3.74 \]

### Step 6: Identify integral values of \( k \)
The integral values of \( k \) within the interval \( (-1.07, 3.74) \) are:
\[ -1, 0, 1, 2, 3 \]
Thus, the integral values of \( k \) are \( -1, 0, 1, 2, 3 \).

### Step 7: Count the integral values
The total number of integral values of \( k \) is:
\[ 5 \]

### Final Answer
The number of integral values of \( k \) is \( 5 \).