Question

If \( \alpha \) and \( \beta \) are the zeros of the polynomial \( x^2 + 5x + 8 \), then the value of \( \alpha^2 + \beta^2 + 2\alpha\beta \) is:

• A. \( 25 \)
• B. \( 5 \)
• C. \( 8 \)
• D. \( 64 \)

Hint:

For any quadratic equation \( ax^2 + bx + c = 0 \): \[ \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}. \]

Answer 1

The Correct Option is A

Using the identity:
\[ \alpha^2 + \beta^2 + 2\alpha\beta = (\alpha + \beta)^2. \] From Vieta’s formulas:
\[ \alpha + \beta = -\frac{5}{1} = -5. \] \[ \alpha^2 + \beta^2 + 2\alpha\beta = (-5)^2 = 25. \]