Intersection between two parabolas

When we want to find the intersection points between two parabolas, we equate their equations – according to the method of comparing quadratic equations.
Note that we compare the equations only when $Y$ is isolated in the same way in both equations.
After equating, we will get one equation with only one variable.
We find this variable and substitute the value of the variable we obtained into one of the original equations to reach $Y$.

Note –
Sometimes, we encounter parabolas that never intersect, and therefore there will be no solution to the new equation we obtain.
For example, in these parabolas:
*Relevant illustration in Word file*

Sometimes, we encounter parabolas that intersect at only one point, and therefore we find only one pair of $X$ and $Y$ in the system of equations.
For example, in these parabolas:

*Relevant illustration in the Word file*

Sometimes, we encounter parabolas that intersect at two points, and therefore we find two pairs of $X$ and $Y$ in the system of equations.
For example, in these parabolas:
*Relevant illustration in a Word file*