Question

If \( \alpha \) and \( \beta \) are the roots of \( x^2 - ax + b^2 = 0 \), then \( \alpha^2 + \beta^2 \) is equal to:

• A. \( a^2 + 2b^2 \)
• B. \( a^2 - 2b^2 \)
• C. \( a^2 - 2b \)
• D. \( a^2 + 2b \)
• E. \( a^2 - b^2 \)

Hint:

Always look at the constant term carefully. Here it was $b^2$, not $b$, so the product is $b^2$.

Answer 1

The Correct Option is B

Concept: Using Vieta's formulas for \( x^2 - Sx + P = 0 \):
• Sum of roots \( \alpha + \beta = S \)
• Product of roots \( \alpha\beta = P \) The sum of squares is given by \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \).

Step 1: Identify the sum and product from the equation.
Equation: \( x^2 - ax + b^2 = 0 \) Sum \( (\alpha + \beta) = a \) Product \( (\alpha\beta) = b^2 \)

Step 2: Substitute into the algebraic identity.
\[ \alpha^2 + \beta^2 = (a)^2 - 2(b^2) \] \[ \alpha^2 + \beta^2 = a^2 - 2b^2 \]