Question

If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 - 8x + 5 = 0 \), then the value of \( \alpha^2 + \beta^2 \) is:

• A. 44
• B. 54
• C. 74
• D. 64

Hint:

Use the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \) to calculate \( \alpha^2 + \beta^2 \) from the sum and product of the roots.

Answer 1

The Correct Option is B

From the given quadratic equation \( x^2 - 8x + 5 = 0 \), the sum and product of the roots \( \alpha \) and \( \beta \) are: \[ \alpha + \beta = -\frac{-8}{1} = 8, \quad \alpha \beta = \frac{5}{1} = 5. \] We need to find \( \alpha^2 + \beta^2 \). Using the identity: \[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta, \] we substitute the values of \( \alpha + \beta \) and \( \alpha \beta \): \[ \alpha^2 + \beta^2 = 8^2 - 2 \times 5 = 64 - 10 = 54. \] Thus, the value of \( \alpha^2 + \beta^2 \) is \( \boxed{54} \).