Quadratic graph / standard form

Question

If I draw a graph of the quadratic $x^2-9=0$, I have a parabola with roots $x=3$ and $x=-3$ and a vertex of $(0,-9)$ with the parabola opening upwards as $a$ is positive in the standard quadratic form. If the original quadratic is given in non-standard form, $x^2=9$, then we can rearrange this by subtracting $x^2$ from both sides which yields $0=-x^2+9$. This quadratic will give the same roots but inverted. When rearranging a quadratic that is given in non standard form, is there a convention to follow in terms of which side of the equation we collect like terms? Thanks.

Answer 1

I would say no there is no standard rearrangement of the equation and any choice would be a personal preference.

The quadratic formula is stated for $ax^2+bx+c=0$ as $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and so doesn't really care about whether $a$ is negative. The formula only notices the quantity $\Delta=b^2-4ac$ to decide the nature of the roots.

There are many quadratic graphs that have the roots $x=3$ and $x= -3$ so you'd need further criterion to decide amongst them.

For example $y=2x^2-18$ has the same roots but a $y$ intercept of $-18$.

I'd also mention that an equation is different from a graph.

The equation $x^2-9=0$ can be seen as the intersection of the graphs of $y=x^2-9$ and $y=0$