Mathematico | Mathematico

The line $y = 2 x - 2$ intersects the parabola

y = 2 x^{2} + 11 x + c

at two distinct points. One of those points is $(- 5, - 12)$ and the other is $(p, q)$ .

Find the value of

128 c (p^{3} + q^{3})

Hint

$(- 5, - 12)$ lies on the parabola, so it satisfies its equation. Substitute the point into the equation to find $c$ .

Solution

Given that $(- 5, - 12)$ lies on the parabola, we know that

\begin{aligned} - 12 & = 2 (- 5)^{2} + 11 (- 5) + c \\ - 12 & = 50 - 55 + c \\ c & = - 7 \end{aligned}

so the equation of the parabola is

y = 2 x^{2} + 11 x - 7

The points of intersection are the solutions of

\begin{aligned} 2 x - 2 & = 2 x^{2} + 11 x - 7 \\ 2 x^{2} + 9 x - 5 & = 0 \\ (2 x - 1) (x + 5) & = 0 \\ x & \in {\frac{1}{2}, - 5} \end{aligned}

So $p = \frac{1}{2}$ and

\begin{aligned} q & = 2 p - 2 \\ = 1 - 2 \\ = - 1 \end{aligned}