Is the graph of this problem statement incorrect?

Question

Here is the problem statement:

According to the parabola in the figure of the function $f(x) = ax^2 + bx + c$, it is given that $a + b + c = 0$. The points $A$ and $B$ are the x-intersects, and the point $C$ is the y-intersect of the parabola. Let the origin be the point $O$, and determine the area of the triangle $COA$.

Here are my solving steps and why I think the plot doesn't seem right.

1. From the figure, it is apparent that $C = 3$, which leaves me with $a + b = -3$, from which I can write $b = - a - 3$.
2. I find the roots of the parabola, which turn out to be $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Substituting $b = - a - 3$ and $c = 3$ the roots are $x_1 = 1$ and $x_2 = \frac{3}{a}$.

However, since the value from under the square root on step 2 above is a positive value, I can assume that $x_1$ > $x_2$, simply because I used $+$ of the $\pm$ for $x_1$, and the $-$ for $x_2$. But the problem's answer accepts 1 to be the x coordinate of the point $A$ and proceeds on calculating the area from there with $\frac{3\times1}{2}=1.5$. I'm intending to appeal for the problem, but can't make sure whether or not I'm missing something.

Answer 1

In solving the quadratic equation, x= $\frac{(a+3)\pm\sqrt{(a+3)^2-12a}}{2a}$ =$\frac{(a+3)\pm|a-3|}{2a}$

1. when a$\geq$3, $x_1$=1>$x_2$=$\frac{3}{a}$;x2 is the smaller root. In this case, the area of $\Delta$ AOC has many values pending on value of a.

2. when a$\leq$3, $x_1$=1<$x_2$=$\frac{3}{a}$;1 is the small root. In this case, the area of $\Delta$ AOC is $\frac{3}{2}$.

Hope this clarify the picture.

Answer 2

We have $a+b+c=f(1)=0$, hence clearly $1$ is a root. The question is whether it is the smallest root.

Also $f(0)=c=3$.

From vieta's formula, $\frac{c}{a}=1\cdot x_2$, that the other root is $\frac{c}{a}=\frac3a$.

Hence if $\frac3a > 1$, then the answer is $\frac32$.

However, it is also possible that $\frac3a < 1$ as well. Then the answer is $\frac{3}{2}\cdot \frac{3}{a}=\frac{9}{2a}$ which is a function of $a$.

For example, consider the graph of $$f(x)=6x^2-9x+3=3(2x^2-3x+1)=3(2x-1)(x-1).$$