The set of possible values of `lambda` for which `x^2-(lambda^2-5 lambda+5)x+(2 lambda^2-3lambda-4)=0` has roots whose sum and product are both less than 1 is

Options

A. `(-1, 5//2)`
B. `(1, 4)`
C. `[1, 5//2]`
D. `(1, 5//2)`

Answer

To solve the problem, we need to find the set of possible values of \( \lambda \) for which the quadratic equation

\[ x^2 - (\lambda^2 - 5\lambda + 5)x + (2\lambda^2 - 3\lambda - 4) = 0 \]

has roots whose sum and product are both less than 1.

### Step 1: Identify the sum and product of the roots

For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of the roots \( \alpha + \beta \) is given by \( -\frac{b}{a} \) and the product of the roots \( \alpha \beta \) is given by \( \frac{c}{a} \).

In our case:
- \( a = 1 \)
- \( b = -(\lambda^2 - 5\lambda + 5) \)
- \( c = 2\lambda^2 - 3\lambda - 4 \)

Thus, we can express the sum and product of the roots as follows:

\[ \alpha + \beta = \lambda^2 - 5\lambda + 5 \]
\[ \alpha \beta = 2\lambda^2 - 3\lambda - 4 \]

### Step 2: Set conditions for the sum of roots

We want the sum of the roots to be less than 1:

\[ \lambda^2 - 5\lambda + 5 < 1 \]

Rearranging this gives:

\[ \lambda^2 - 5\lambda + 4 < 0 \]

### Step 3: Factor the quadratic inequality

We can factor the quadratic:

\[ (\lambda - 1)(\lambda - 4) < 0 \]

### Step 4: Determine the intervals

The roots of the equation \( (\lambda - 1)(\lambda - 4) = 0 \) are \( \lambda = 1 \) and \( \lambda = 4 \). The intervals to test are:

- \( (-\infty, 1) \)
- \( (1, 4) \)
- \( (4, \infty) \)

Testing these intervals, we find that \( (\lambda - 1)(\lambda - 4) < 0 \) holds for:

\[ 1 < \lambda < 4 \]

### Step 5: Set conditions for the product of roots

Next, we want the product of the roots to also be less than 1:

\[ 2\lambda^2 - 3\lambda - 4 < 1 \]

Rearranging gives:

\[ 2\lambda^2 - 3\lambda - 5 < 0 \]

### Step 6: Factor the quadratic inequality

We can factor this quadratic:

\[ (2\lambda + 1)(\lambda - 5/2) < 0 \]

### Step 7: Determine the intervals

The roots of the equation \( (2\lambda + 1)(\lambda - 5/2) = 0 \) are \( \lambda = -1/2 \) and \( \lambda = 5/2 \). The intervals to test are:

- \( (-\infty, -1/2) \)
- \( (-1/2, 5/2) \)
- \( (5/2, \infty) \)

Testing these intervals, we find that \( (2\lambda + 1)(\lambda - 5/2) < 0 \) holds for:

\[ -1/2 < \lambda < 5/2 \]

### Step 8: Find the intersection of the intervals

Now we have two conditions:

1. \( 1 < \lambda < 4 \)
2. \( -1/2 < \lambda < 5/2 \)

The intersection of these intervals is:

\[ 1 < \lambda < 5/2 \]

### Final Result

Thus, the set of possible values of \( \lambda \) is:

\[ \lambda \in (1, 5/2) \]