Mathematico | Mathematico

The parabola with equation

y = 3 x^{2} + 9 x - 30

intersects the $x$ -axis at the points $A$ and $B$ .

Let the length of $A B$ be $ℓ$ , and the coordinates of the minimum point be $(x, y)$ .

Find the value of

16 x y ℓ

Hint

Find the coordinates of $A$ and $B$ by solving $3 x^{2} + 9 x - 30 = 0.$

Solution

Parabola

To find $A$ and $B$ , we need the roots:

\begin{aligned} 3 x^{2} + 9 x - 30 & = 0 \\ x^{2} + 3 x - 10 & = 0 \\ (x + 5) (x - 2) & = 0 \\ x & \in {- 5, 2} \end{aligned}

So $A (- 5, 0)$ and $B (2, 0)$ and $ℓ = 7$ .

The $x$ coordinate of the minimum point is $\frac{- 5 + 2}{2} = - 1.5$ .

The $y$ coordinate is

\begin{aligned} y & = 3 (- 1.5)^{2} + 9 (- 1.5) - 30 \\ = - 36.75 \end{aligned}

So we have

\begin{aligned} 16 x y ℓ & = 16 (- 1.5) (- 36.75) (7) \\ = 6174 \end{aligned}