The value of a for which the sum of the square of the roots of the equation `x^(2)-(a-2)x-a+1=0` is least, is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the value of \( a \) for which the sum of the squares of the roots of the equation

\[ x^2 - (a-2)x - (a-1) = 0 \]

is minimized, we can follow these steps:

### Step 1: Identify coefficients
The given quadratic equation can be expressed in the standard form \( Ax^2 + Bx + C = 0 \), where:
- \( A = 1 \)
- \( B = -(a-2) \)
- \( C = -(a-1) \)

### Step 2: Calculate the sum of the roots
The sum of the roots \( \alpha + \beta \) of the quadratic equation can be calculated using the formula:
\[ \alpha + \beta = -\frac{B}{A} = -\frac{-(a-2)}{1} = a - 2 \]

### Step 3: Calculate the product of the roots
The product of the roots \( \alpha \beta \) can be calculated using the formula:
\[ \alpha \beta = \frac{C}{A} = \frac{-(a-1)}{1} = -a + 1 \]

### Step 4: Calculate the sum of the squares of the roots
The sum of the squares of the roots \( \alpha^2 + \beta^2 \) can be expressed in terms of the sum and product of the roots:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we found:
\[ \alpha^2 + \beta^2 = (a - 2)^2 - 2(-a + 1) \]
\[ = (a - 2)^2 + 2a - 2 \]

### Step 5: Expand and simplify
Now, expand \( (a - 2)^2 \):
\[ (a - 2)^2 = a^2 - 4a + 4 \]
So we have:
\[ \alpha^2 + \beta^2 = a^2 - 4a + 4 + 2a - 2 \]
\[ = a^2 - 2a + 2 \]

### Step 6: Minimize the quadratic expression
To find the value of \( a \) that minimizes \( a^2 - 2a + 2 \), we can use the vertex formula for a quadratic equation \( ax^2 + bx + c \), where the vertex occurs at \( a = -\frac{b}{2a} \):
Here, \( a = 1 \) and \( b = -2 \):
\[ a = -\frac{-2}{2 \cdot 1} = 1 \]

### Step 7: Conclusion
Thus, the value of \( a \) for which the sum of the squares of the roots is minimized is:
\[ \boxed{1} \]