For the quadratic equation 3x2 − 4x + p = 0, if alpha and beta are the roots, show that alpha2 + beta2 = (16 − 6p)/(9).

Step 1: Solve Question 1.

a) Given the roots of the equation $3x^2 - 4x + p = 0$ are $\alpha$ and $\beta$.
For a quadratic equation $ax^2 + bx + c = 0$, the sum of roots is $\alpha + \beta = -\frac{b}{a}$ and the product of roots is $\alpha\beta = \frac{c}{a}$.
For $3x^2 - 4x + p = 0$:

$$\alpha + \beta = -\frac{-4}{3} = \frac{4}{3}$$ $$\alpha\beta = \frac{p}{3}$$

(i) Show that $\alpha^2 + \