If both the roots of the equation `x^(2)-12kx+k^(2)+k-5=0` are less than 5, then k lies in the inverval

Options

A. `(-oo,4)`
B. `[4,5]`
C. `[5,6]`
D. `(6,oo)`

Answer

To solve the problem, we need to analyze the quadratic equation given by:

\[ x^2 - 12kx + (k^2 + k - 5) = 0 \]

We want to find the interval for \( k \) such that both roots of this equation are less than 5.

### Step 1: Use the Sum of Roots Condition

The sum of the roots \( \alpha + \beta \) of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[ \alpha + \beta = -\frac{b}{a} \]

In our case, \( a = 1 \) and \( b = -12k \). Therefore, the sum of the roots is:

\[ \alpha + \beta = 12k \]

Since we want both roots to be less than 5, we have:

\[ \alpha + \beta < 10 \]

Substituting the expression for the sum of the roots, we get:

\[ 12k < 10 \]

### Step 2: Solve for \( k \)

Now, we can solve the inequality:

\[ k < \frac{10}{12} = \frac{5}{6} \]

### Step 3: Use the Product of Roots Condition

Next, we consider the product of the roots \( \alpha \beta \), which is given by:

\[ \alpha \beta = \frac{c}{a} = k^2 + k - 5 \]

For both roots to be less than 5, the product of the roots must also be positive:

\[ \alpha \beta > 0 \]

This means:

\[ k^2 + k - 5 > 0 \]

### Step 4: Find the Roots of the Quadratic Inequality

To solve the inequality \( k^2 + k - 5 > 0 \), we first find the roots of the equation \( k^2 + k - 5 = 0 \) using the quadratic formula:

\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-5)}}{2 \cdot 1} \]

Calculating the discriminant:

\[ 1 + 20 = 21 \]

Thus, the roots are:

\[ k = \frac{-1 \pm \sqrt{21}}{2} \]

Let’s denote the roots as \( k_1 \) and \( k_2 \):

\[ k_1 = \frac{-1 - \sqrt{21}}{2}, \quad k_2 = \frac{-1 + \sqrt{21}}{2} \]

### Step 5: Determine the Intervals

The quadratic \( k^2 + k - 5 \) opens upwards (since the coefficient of \( k^2 \) is positive). Therefore, the inequality \( k^2 + k - 5 > 0 \) holds outside the interval \( (k_1, k_2) \).

### Step 6: Combine the Results

We have two conditions for \( k \):

1. \( k < \frac{5}{6} \)
2. \( k < k_1 \) or \( k > k_2 \)

Now we need to find the intersection of these intervals.

### Conclusion

The final interval for \( k \) such that both roots of the equation are less than 5 is:

\[ k \in (-\infty, k_1) \cup (k_2, \frac{5}{6}) \]