Question

If \(\alpha, \beta\) are the roots of the equation \(2x^2 - 3x + 1 = 0\), then the value of \(\alpha^3 + \beta^3\) is

• A. \(\frac{9}{8}\)
• B. 8
• C. \(\frac{8}{9}\)
• D. 16

Hint:

Use the identity for cubes of roots when solving for expressions involving the sum of cubes of roots of a quadratic equation.

Answer 1

The Correct Option is C

We use the identity: \[ \alpha^3 + \beta^3 = (\alpha + \beta)\left[(\alpha + \beta)^2 - 3\alpha\beta\right] \] From the equation, we have: \[ \alpha + \beta = \frac{3}{2} \quad \text{and} \quad \alpha\beta = \frac{1}{2} \] Substituting into the identity: \[ \alpha^3 + \beta^3 = \frac{3}{2} \times \left[\left(\frac{3}{2}\right)^2 - 3 \times \frac{1}{2}\right] = \frac{8}{9} \] Therefore, the answer is \(\frac{8}{9}\).