Question

If \( \alpha \), \( \beta \), and \( \gamma \) are the roots of the equation \( x^3 - ax^2 + bx - c = 0 \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is:

• A. \( \frac{b^2 - 3ac}{c^2} \)
• B. \( \frac{b^2 - ac}{c^2} \)
• C. \( \frac{b^2 - 2ac}{c^2} \)
• D. \( \frac{b^2 - 4ac}{c^2} \)

Hint:

To find expressions involving the squares of the roots, use identities and relationships between the roots and coefficients.

Answer 1

The Correct Option is C

We are given the cubic equation: \[ x^3 - ax^2 + bx - c = 0 \] The sum and product of the roots \( \alpha \), \( \beta \), and \( \gamma \) are related to the coefficients as follows (using Vieta's formulas): \[ \alpha + \beta + \gamma = a, \quad \alpha\beta + \beta\gamma + \gamma\alpha = b, \quad \alpha\beta\gamma = c \] We are asked to find \( \alpha^2 + \beta^2 + \gamma^2 \). Using the identity: \[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \] Substitute the known values: \[ \alpha^2 + \beta^2 + \gamma^2 = a^2 - 2b \] Thus, the correct answer is \( \frac{b^2 - 2ac}{c^2} \).