Key Concepts and Formulas

• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$, we have $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$.
• Sum of Squares Identity: $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$
• Completing the Square: A technique to rewrite a quadratic expression in the form $a(x-h)^2 + k$ to easily find its minimum or maximum value.

Step-by-Step Solution

Step 1: Rewrite the equation in standard form

The given equation is $x^{2}+(3-a) x+1=2 a$. To apply Vieta's formulas, we need to rewrite it in the standard quadratic form $Ax^2 + Bx + C = 0$. Subtracting $2a$ from both sides, we get: $x^{2}+(3-a) x+(1-2 a) = 0$

Step 2: Identify coefficients and apply Vieta's formulas

Now, we can identify the coefficients:

• $A = 1$
• $B = (3-a)$
• $C = (1-2a)$

Let $\alpha$ and $\beta$ be the roots of the equation. Applying Vieta's formulas:

• Sum of roots: $\alpha + \beta = -\frac{B}{A} = -\frac{3-a}{1} = a-3$
• Product of roots: $\alpha \beta = \frac{C}{A} = \frac{1-2a}{1} = 1-2a$

Step 3: Express the sum of squares of roots in terms of 'a'

We want to find the minimum value of $\alpha^2 + \beta^2$. Using the sum of squares identity: $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$ Substitute the expressions for $\alpha + \beta$ and $\alpha\beta$ in terms of '$a$': $\alpha^2 + \beta^2 = (a-3)^2 - 2(1-2a)$

Step 4: Simplify the expression and complete the square

Expand and simplify the expression: $\alpha^2 + \beta^2 = (a^2 - 6a + 9) - 2 + 4a$ $\alpha^2 + \beta^2 = a^2 - 2a + 7$ Now, complete the square: $\alpha^2 + \beta^2 = (a^2 - 2a + 1) + 6$ $\alpha^2 + \beta^2 = (a-1)^2 + 6$

Step 5: Find the minimum value

The expression $(a-1)^2$ is always non-negative. Its minimum value is 0, which occurs when $a = 1$. Therefore, the minimum value of $\alpha^2 + \beta^2$ is: $(1-1)^2 + 6 = 0 + 6 = 6$

Common Mistakes & Tips

• Mistake: Forgetting to rearrange the equation into the standard form before applying Vieta's formulas.
• Mistake: Sign errors when applying Vieta's formulas, especially for the sum of roots.
• Tip: Completing the square is a powerful technique for finding the minimum or maximum value of a quadratic expression.

Summary

By rewriting the given quadratic equation in standard form and applying Vieta's formulas, we expressed the sum of the squares of the roots as a quadratic function of $a$: $a^2 - 2a + 7$. Completing the square, we found that this expression is equivalent to $(a-1)^2 + 6$. The minimum value of this expression occurs when $a=1$, and the minimum value is 6. This corresponds to option (C).

Final Answer

The final answer is \boxed{6}, which corresponds to option (C).