1.25: Past-paper question 10

Given that $\displaystyle f(x)$ can be expressed in the form $\displaystyle A - B(x + C)^2$ where $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ are positive constants

(a) find the value of $\displaystyle A$, the value of $\displaystyle B$ and the value of $\displaystyle C$

(b) Hence write down the maximum value of $\displaystyle f(x)$

The equation $\displaystyle f(x) = 0$ has roots $\displaystyle \alpha$ and $\displaystyle \beta$

Without solving the equation $\displaystyle f(x) = 0$

(c) form a quadratic equation, with integer coefficients, that has roots $\displaystyle \frac{3\alpha}{\beta}$ and $\displaystyle \frac{3\beta}{\alpha}$

(d) Show that $\displaystyle (x + y)^3 = x^3 + y^3 + 3xy(x + y)$

$\displaystyle g(x) = 3x^2 + qx + r$ where $\displaystyle q$ and $\displaystyle r$ are constants

The equation $\displaystyle g(x) = 0$ has roots $\displaystyle \alpha^2 - \beta$ and $\displaystyle \beta^2 - \alpha$ where $\displaystyle \alpha$ and $\displaystyle \beta$ are the roots of the equation $\displaystyle f(x) = 0$

(e) Using your answer to part (d), find in simplified exact form, the value of $\displaystyle q$ and the value of $\displaystyle r$