Question 1: Roots of a Quadratic Equation and Inequality

Here are the solutions to the questions:

Question 1: Roots of a Quadratic Equation and Inequality

a) Given the equation $3x^2 - 4x + 2 = 0$.
The roots are $\alpha$ and $\beta$.
From Vieta's formulas, for a quadratic equation $ax^2 + bx + c = 0$:
Sum of roots: $\alpha + \beta = -\frac{b}{a} = -\frac{(-4)}{3} = \frac{4}{3}$
Product of roots: $\alpha\beta = \frac{c}{a} = \frac{2}{3}$

We need to show that $\alpha^3 + \beta^3 = -\frac{20}{27}$.