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 find the values of \( \lambda \) such that the sum of the squares of the roots of the quadratic equation \( x^2 + (3 - \lambda)x + 2 = \lambda \) has the least value, we can follow these steps:

### Step 1: Rewrite the Quadratic Equation
The given quadratic equation can be rewritten as:
\[ x^2 + (3 - \lambda)x + (2 - \lambda) = 0 \]

### Step 2: Identify the Roots
Let the roots of the quadratic be \( \alpha \) and \( \beta \). By Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -(3 - \lambda) = \lambda - 3 \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = 2 - \lambda \)

### Step 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 we found:
\[ \alpha^2 + \beta^2 = (\lambda - 3)^2 - 2(2 - \lambda) \]

### Step 4: Simplify the Expression
Now, let's simplify the expression:
\[ \alpha^2 + \beta^2 = (\lambda - 3)^2 - 4 + 2\lambda \]
Expanding \( (\lambda - 3)^2 \):
\[ = \lambda^2 - 6\lambda + 9 - 4 + 2\lambda \]
Combining like terms:
\[ = \lambda^2 - 4\lambda + 5 \]

### Step 5: Find the Minimum Value
To find the minimum value of \( \alpha^2 + \beta^2 \), we need to differentiate the expression with respect to \( \lambda \):
\[ f(\lambda) = \lambda^2 - 4\lambda + 5 \]
Differentiating:
\[ f'(\lambda) = 2\lambda - 4 \]
Setting the derivative to zero to find critical points:
\[ 2\lambda - 4 = 0 \implies \lambda = 2 \]

### Step 6: Verify the Nature of the Critical Point
To determine if this critical point is a minimum, we check the second derivative:
\[ f''(\lambda) = 2 \]
Since \( f''(\lambda) > 0 \), the function has a minimum at \( \lambda = 2 \).

### Conclusion
Thus, the value of \( \lambda \) such that the sum of the squares of the roots has the least value is:
\[ \boxed{2} \]