The value of k for which both the roots of the equation `4x^(2)-20kx+(25k^(2)+15k-66)=0` are less than 2, lies in

Options

A. (4/5,2)
B. (2,0)
C. (-1,-4/5)
D. `(-oo, -1)`

Answer

To find the value of \( k \) for which both the roots of the equation

\[ 4x^2 - 20kx + (25k^2 + 15k - 66) = 0 \]

are less than 2, we will follow these steps:

### Step 1: Sum of the Roots
The sum of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:

\[ \text{Sum of roots} = -\frac{b}{a} \]

Here, \( a = 4 \) and \( b = -20k \). Therefore, the sum of the roots is:

\[ \text{Sum of roots} = -\frac{-20k}{4} = 5k \]

### Step 2: Condition for Roots to be Less than 2
For both roots to be less than 2, the sum of the roots must be less than 4 (since \( 2 + 2 = 4 \)). Thus, we have:

\[ 5k < 4 \]

Dividing both sides by 5 gives:

\[ k < \frac{4}{5} \]

### Step 3: Product of the Roots
Next, we need to ensure that the product of the roots is positive. The product of the roots is given by:

\[ \text{Product of roots} = \frac{c}{a} \]

Here, \( c = 25k^2 + 15k - 66 \). Therefore, the product of the roots is:

\[ \text{Product of roots} = \frac{25k^2 + 15k - 66}{4} \]

For both roots to be less than 2, we need:

\[ 25k^2 + 15k - 66 > 0 \]

### Step 4: Finding the Roots of the Quadratic Inequality
To find the values of \( k \) that satisfy \( 25k^2 + 15k - 66 = 0 \), we will use the quadratic formula:

\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting \( a = 25 \), \( b = 15 \), and \( c = -66 \):

\[ k = \frac{-15 \pm \sqrt{15^2 - 4 \cdot 25 \cdot (-66)}}{2 \cdot 25} \]

Calculating the discriminant:

\[ 15^2 = 225 \]
\[ -4 \cdot 25 \cdot (-66) = 6600 \]
\[ \text{Discriminant} = 225 + 6600 = 6825 \]

Now substituting back into the formula:

\[ k = \frac{-15 \pm \sqrt{6825}}{50} \]

Calculating \( \sqrt{6825} \):

\[ \sqrt{6825} \approx 82.6 \quad (\text{exact value can be calculated if needed}) \]

Thus,

\[ k = \frac{-15 \pm 82.6}{50} \]

Calculating the two possible values for \( k \):

1. \( k_1 = \frac{-15 + 82.6}{50} \approx 1.35 \)
2. \( k_2 = \frac{-15 - 82.6}{50} \approx -1.95 \)

### Step 5: Analyzing the Intervals
The roots of the quadratic inequality \( 25k^2 + 15k - 66 > 0 \) will give us intervals. The critical points are \( k \approx -1.95 \) and \( k \approx 1.35 \).

To determine where the inequality holds, we test intervals:

1. \( k < -1.95 \)
2. \( -1.95 < k < 1.35 \)
3. \( k > 1.35 \)

Testing a value in each interval, we find:

- For \( k < -1.95 \), the product is positive.
- For \( -1.95 < k < 1.35 \), the product is negative.
- For \( k > 1.35 \), the product is positive.

Thus, the intervals where \( 25k^2 + 15k - 66 > 0 \) are \( (-\infty, -1.95) \) and \( (1.35, \infty) \).

### Step 6: Final Intersection
Now we combine the conditions:

1. From the sum of roots: \( k < \frac{4}{5} \)
2. From the product of roots: \( k \in (-\infty, -1.95) \cup (1.35, \infty) \)

The only interval that satisfies both conditions is:

\[ k \in (-\infty, -1.95) \]

### Conclusion
The value of \( k \) for which both roots of the equation are less than 2 lies in the interval:

\[ (-\infty, -1.95) \]