Mathematico | Mathematico

The graph of the parabola

y = a x^{2} + b x + c

is sketched below:

Find the value of $a b c$ .

Hint

Since the $y$ intercept is $2 a$ , we can replace $c$ with $2 a$

Hint

Since $a$ is a root, we know that

a (a)^{2} + b (a) + c = 0

Write something similar for $2 a$ and try to solve the equations (you can subtract them).

Solution

By considering the $y$ intercept, we know that $c = 2 a$ , so

y = a x^{2} + b x + 2 a

The roots are $x = a$ and $x = 2 a$ so

\begin{aligned} a (a)^{2} + b (a) + 2 a & = 0 \\ a^{3} + a b + 2 a & = 0 & (i) \end{aligned}

and

\begin{aligned} a (2 a)^{2} + b (2 a) + 2 a & = 0 \\ 4 a^{3} + 2 a b + 2 a & = 0 & (i i) \end{aligned}

Solving simultaneously to eliminate $b$ , we have

\begin{aligned} (i i) - 2 (i) : & 2 a^{3} - 2 a & = 0 \\ 2 a (a^{2} - 1) & = 0 \\ 2 a (a - 1) (a + 1) & = 0 \\ a & \in {0, 1, - 1} \end{aligned}

Clearly $a \neq 0$ , otherwise this is not a parabola. Also, the parabola is convex, so $a > 0$ . The only possibility is $a = 1$ .

We have already said that $c = 2 a$ , so $c = 2$ .

The equation of the parabola is

\begin{aligned} y & = x^{2} + b x + 2 \end{aligned}

It has a root at $x = 1$ , so

\begin{aligned} 0 & = 1 + b + 2 \\ b & = - 3 \end{aligned}