If G and L are the greatest and least values of the expression`(2x^(2)-3x+2)/(2x^(2)+3x+2), x epsilonR` respectively. If `L^(2)ltlamdaltG^(2), lamda epsilon N` the sum of all values of `lamda` is

Options

A. 1035
B. 1081
C. 1225
D. 1176

Explanation

To solve the problem, we need to find the greatest (G) and least (L) values of the expression \( y = \frac{2x^2 - 3x + 2}{2x^2 + 3x + 2} \) and then determine the sum of all natural numbers \( \lambda \) that satisfy \( L^2 < \lambda < G^2 \).

### Step 1: Define the expression
Let:
\[ y = \frac{2x^2 - 3x + 2}{2x^2 + 3x + 2} \]

### Step 2: Rearrange the equation
Multiply both sides by the denominator (assuming it is not zero):
\[ y(2x^2 + 3x + 2) = 2x^2 - 3x + 2 \]
Rearranging gives:
\[ 2x^2y + 3xy + 2y = 2x^2 - 3x + 2 \]
This can be rearranged to:
\[ (2y - 2)x^2 + (3y + 3)x + (2y - 2) = 0 \]

### Step 3: Apply the discriminant condition
For \( x \) to have real solutions, the discriminant \( D \) of the quadratic must be non-negative:
\[ D = b^2 - 4ac \geq 0 \]
Here, \( a = 2y - 2 \), \( b = 3y + 3 \), and \( c = 2y - 2 \). Thus:
\[ D = (3y + 3)^2 - 4(2y - 2)(2y - 2) \geq 0 \]

### Step 4: Simplify the discriminant
Calculating \( D \):
\[ D = (3y + 3)^2 - 4(2y - 2)^2 \]
Expanding:
\[ D = 9y^2 + 18y + 9 - 4(4y^2 - 8y + 4) \]
\[ = 9y^2 + 18y + 9 - (16y^2 - 32y + 16) \]
\[ = -7y^2 + 50y - 7 \geq 0 \]

### Step 5: Solve the quadratic inequality
Rearranging gives:
\[ 7y^2 - 50y + 7 \leq 0 \]
Finding the roots using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{50 \pm \sqrt{(-50)^2 - 4 \cdot 7 \cdot 7}}{2 \cdot 7} \]
Calculating the discriminant:
\[ = \sqrt{2500 - 196} = \sqrt{2304} = 48 \]
Thus, the roots are:
\[ y = \frac{50 \pm 48}{14} \]
Calculating the roots:
\[ y_1 = \frac{98}{14} = 7, \quad y_2 = \frac{2}{14} = \frac{1}{7} \]

### Step 6: Determine the range of \( y \)
The inequality \( 7y^2 - 50y + 7 \leq 0 \) gives the range:
\[ \frac{1}{7} \leq y \leq 7 \]
Thus, \( L = \frac{1}{7} \) and \( G = 7 \).

### Step 7: Calculate \( L^2 \) and \( G^2 \)
Calculating:
\[ L^2 = \left(\frac{1}{7}\right)^2 = \frac{1}{49}, \quad G^2 = 7^2 = 49 \]

### Step 8: Find \( \lambda \) values
We need to find \( \lambda \) such that:
\[ \frac{1}{49} < \lambda < 49 \]
The natural numbers satisfying this inequality are \( 1, 2, \ldots, 48 \).

### Step 9: Calculate the sum of \( \lambda \)
The sum of the first 48 natural numbers:
\[ \text{Sum} = \frac{n(n + 1)}{2} = \frac{48 \cdot 49}{2} = 1176 \]

### Final Answer
The sum of all values of \( \lambda \) is:
\[ \boxed{1176} \]