Key Concepts and Formulas

• Vieta's Formulas: For a quadratic equation $Ax^2 + Bx + C = 0$ with roots $\alpha$ and $\beta$, the sum of the roots is $\alpha + \beta = -\frac{B}{A}$ and the product of the roots is $\alpha \beta = \frac{C}{A}$.
• Sum of Squares of Roots: $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$
• Vertex of a Parabola: The vertex of the parabola $f(x) = ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$. If $a > 0$, the vertex is a minimum.

Step-by-Step Solution

Step 1: Identify Coefficients and Apply Vieta's Formulas

The given quadratic equation is $x^2 - (a - 2)x - a - 1 = 0$. We identify the coefficients as:

• $A = 1$
• $B = -(a - 2)$
• $C = -(a + 1)$

Using Vieta's formulas, we find the sum and product of the roots $\alpha$ and $\beta$:

• Sum of roots: $\alpha + \beta = -\frac{B}{A} = -\frac{-(a - 2)}{1} = a - 2$
• Product of roots: $\alpha \beta = \frac{C}{A} = \frac{-(a + 1)}{1} = -(a + 1)$

These expressions relate the sum and product of the roots to the parameter '$a$'.

Step 2: Express the Sum of Squares of Roots in Terms of '$a$'

We want to minimize $\alpha^2 + \beta^2$. Using the identity $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$, we substitute the expressions for $\alpha + \beta$ and $\alpha \beta$ from Step 1: $\alpha^2 + \beta^2 = (a - 2)^2 - 2(-(a + 1))$ $\alpha^2 + \beta^2 = (a^2 - 4a + 4) + 2(a + 1)$ $\alpha^2 + \beta^2 = a^2 - 4a + 4 + 2a + 2$ $\alpha^2 + \beta^2 = a^2 - 2a + 6$ Let $S(a) = a^2 - 2a + 6$. This is the expression we want to minimize.

Step 3: Find the Value of '$a$' that Minimizes $S(a)$

$S(a) = a^2 - 2a + 6$ is a quadratic function. Since the coefficient of $a^2$ is positive, it has a minimum value. We can find the value of '$a$' that minimizes $S(a)$ using the vertex formula: $a = -\frac{b}{2A} = -\frac{-2}{2(1)} = \frac{2}{2} = 1$ Alternatively, we can complete the square: $S(a) = a^2 - 2a + 1 + 5 = (a - 1)^2 + 5$ The minimum value of $S(a)$ occurs when $(a - 1)^2 = 0$, which means $a = 1$.

Therefore, the sum of the squares of the roots is minimized when $a = 1$.

Common Mistakes & Tips

• Be careful with signs when applying Vieta's formulas and expanding expressions.
• Remember to find the value of '$a$' that minimizes the sum of squares, not the minimum value itself.
• Double-check algebraic manipulations to avoid errors.

Summary By applying Vieta's formulas, we expressed the sum of the squares of the roots as a quadratic function of '$a$'. Minimizing this quadratic function using the vertex formula, we found that the sum of the squares of the roots is minimized when $a = 1$.

Final Answer The final answer is \boxed{1}, which corresponds to option (A).