Question

If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 + 6x + 5 = 0 \), then the value of \( \alpha^2 + \beta^2 \) is:

• A. \( 30 \)
• B. \( 16 \)
• C. \( 26 \)
• D. \( 20 \)

Hint:

The identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \) is useful for finding root sums without solving for \( \alpha \) and \( \beta \) explicitly.

Answer 1

The Correct Option is C

Step 1: Use the identity for sum of squares \[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \] From the equation \( x^2 + 6x + 5 = 0 \): - Sum of roots: \( \alpha + \beta = -\frac{6}{1} = -6 \) - Product of roots: \( \alpha\beta = \frac{5}{1} = 5 \) Step 2: Compute \( \alpha^2 + \beta^2 \) \[ \alpha^2 + \beta^2 = (-6)^2 - 2(5) \] \[ = 36 - 10 \] \[ = 26 \] Thus, the correct answer is \( 26 \).