The values of `lambda` such that sum of the squares of the roots of the quadratic equation `x^(2) + (3 - lambda) x + 2 = lambda` has the least value is

Options

A. 2
B. `(4)/(9)`
C. `(15)/(8)`
D. 1

Answer

To solve the problem, we need to find the values of \( \lambda \) such that the sum of the squares of the roots of the quadratic equation \( x^2 + (3 - \lambda)x + 2 = \lambda \) is minimized.

### Step-by-Step Solution:

1. **Rewrite the Quadratic Equation:**
Start by rearranging the given equation into standard quadratic form:
\[ x^2 + (3 - \lambda)x + (2 - \lambda) = 0. \]
Here, \( A = 1 \), \( B = 3 - \lambda \), and \( C = 2 - \lambda \).

2. **Sum and Product of Roots:**
Using Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{B}{A} = \lambda - 3 \).
- The product of the roots \( \alpha \beta = \frac{C}{A} = 2 - \lambda \).

3. **Sum of the Squares of the Roots:**
The sum of the squares of the roots can be expressed as:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta. \]
Substituting the values from Vieta's formulas:
\[ \alpha^2 + \beta^2 = (\lambda - 3)^2 - 2(2 - \lambda). \]

4. **Simplifying the Expression:**
Expand and simplify the expression:
\[ \alpha^2 + \beta^2 = (\lambda - 3)^2 - 4 + 2\lambda. \]
Expanding \( (\lambda - 3)^2 \):
\[ = \lambda^2 - 6\lambda + 9 - 4 + 2\lambda = \lambda^2 - 4\lambda + 5. \]

5. **Finding the Minimum Value:**
We need to minimize the quadratic expression \( \lambda^2 - 4\lambda + 5 \). This is a standard quadratic equation in the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = -4 \), and \( c = 5 \).

The vertex of a quadratic \( ax^2 + bx + c \) is given by \( \lambda = -\frac{b}{2a} \):
\[ \lambda = -\frac{-4}{2 \cdot 1} = 2. \]

6. **Calculating the Minimum Value:**
Substitute \( \lambda = 2 \) back into the expression to find the minimum value:
\[ \alpha^2 + \beta^2 = (2)^2 - 4(2) + 5 = 4 - 8 + 5 = 1. \]

### Conclusion:
The value of \( \lambda \) that minimizes the sum of the squares of the roots is:
\[ \boxed{2}. \]