For example, let's consider the quadratic equation $-3x^2 + 6x -2 = 0$.
Multiplying both sides by $-1$, we get the equation $3x^2 - 6x +2 = 0$.
The graph of the above equations are different even though they have the same roots. The graph of the first equation is an inverted parabola while the graph of the second equation is an upright parabola (because of the sign of the coefficient of the $x^2$ term).
If we divide both sides of the original equation with $-3$, we get $x^2 - 2x + 2/3 = 0$. The graph of this equation is different from the first 2 equations. It has different roots as well.
Does this imply that multiplying or dividing both sides of a quadratic equation by a constant changes its characteristics? If so, does it also imply that we should not multiply or divide both sides of a quadratic equation by a constant when we try to find its roots? In general, when are we permitted to multiply or divide both sides of an equation by a constant?
If we are searching the roots of an equation, as: $ax^2+bx+c=0$ and $k\ne 0$ is a constant, than, multiplying both sides by $k$ we have: $$ kax^2+kbx+kc=k(ax^2+bx+c)=k\cdot 0=0 $$ and, since $k\ne 0$, we must have $ax^2+bx+c=0$ and this means that the roots of the two equations are the same. ( note that we can do the same for a polynomial of any degree).
This means that we can multiply an equation by a constant $\ne 0$ and we always find the same roots.
Your observation that the multiplication by a negative constant inverts the parabola, is important when we are solving an inequality. E.g.:
$ax^2+bx+c >0 \quad \iff \quad kax^2+kbx+kc > 0 $ iff $k>0$
but $ax^2+bx+c >0 \quad \iff \quad kax^2+kbx+kc < 0 $ if $k<0$