The value of k for which the sum of the squares of the roots of `2x^(2)-2(k-2)x-k=0` is least is

Options

A. 1
B. `(3)/(2)`
C. 2
D. `(5)/(2)`

Answer

To find the value of \( k \) for which the sum of the squares of the roots of the quadratic equation \( 2x^2 - 2(k-2)x - k = 0 \) is minimized, we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is:
\[ 2x^2 - 2(k-2)x - k = 0 \]
Here, we can identify:
- \( a = 2 \)
- \( b = -2(k-2) \)
- \( c = -k \)

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-2(k-2)}{2} = k - 2 \]
- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{-k}{2} \]

### Step 3: Find the 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:
\[ \alpha^2 + \beta^2 = (k - 2)^2 - 2\left(-\frac{k}{2}\right) \]
\[ = (k - 2)^2 + k \]
Expanding \( (k - 2)^2 \):
\[ = k^2 - 4k + 4 + k = k^2 - 3k + 4 \]

### Step 4: Minimize the quadratic function
We need to minimize the function:
\[ f(k) = k^2 - 3k + 4 \]
This is a quadratic function, and it opens upwards (since the coefficient of \( k^2 \) is positive).

### Step 5: Find the vertex of the quadratic
The vertex of a quadratic function \( ax^2 + bx + c \) occurs at:
\[ k = -\frac{b}{2a} \]
For our function:
- \( a = 1 \)
- \( b = -3 \)

Thus,
\[ k = -\frac{-3}{2 \cdot 1} = \frac{3}{2} \]

### Conclusion
The value of \( k \) for which the sum of the squares of the roots is minimized is:
\[ \boxed{\frac{3}{2}} \]