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` assumes the least value.

Explanation

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

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

assumes the least value, we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the 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 \( \alpha + \beta \) is given by 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 \( \alpha \beta \) is given by:

\[ \alpha \beta = \frac{c}{a} = \frac{-(a + 1)}{1} = -(a + 1) \]

### Step 4: Express 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:

\[ \alpha^2 + \beta^2 = (a - 2)^2 - 2(- (a + 1)) \]

### Step 5: Simplify the expression
Now, let's simplify the expression:

\[ \alpha^2 + \beta^2 = (a - 2)^2 + 2(a + 1) \]

Expanding \( (a - 2)^2 \):

\[ = a^2 - 4a + 4 + 2a + 2 \]

Combining like terms:

\[ = a^2 - 2a + 6 \]

### Step 6: Find the minimum value
The expression \( a^2 - 2a + 6 \) is a quadratic function in \( a \). The minimum value of a quadratic \( ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \). 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 assumes the least value is:

\[ \boxed{1} \]