For a quadratic function, there is one minimum/maximum (the proof requires calculus) and also it is either always convex or concave (prove is also calculus) it is continuous every where, hence, it can have a maximum of 2 roots.
Graph it.
If there is more than 2 roots, by Intermediate Value Theorem, it cannot be convex/concave everywhere. It will HAVE to have two intervals of increasing or decreasing. It can be easily proven that given any quadratic function f(x), if x = x0 is a minimum/maximum, and x=a != x0 is a root, then 2x0-a is also a root. It is still true that a = x0 as 2x0-x0=x0 implying it is the only root.
But the concept of min/max requires Calculus to prove existence.
So, this is Calculus, not algebra.
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The x co-ordinate of a quadratic lies exactly halfway between the two x-intercepts, assuming they exist. Alternatively, the x co-ordinate can be found using the formula -B/(2A), when the function is in the form, y = Axx + Bx + C.
The same number as the highest power of the independent variable.
solutions are the well solution to the problem. X-intercepts are wherever a graph cross the x axis, which are hte solutions when you have to find out what x is, zeros are pretty much the same thing although i think that include y-intercepts as well..... not sure. and roots are the same thing as x-intercepts. so they are all more or less the same thing
If you are looking for the zeros of this function: x = -2 plus or minus 2 X the square root of 5.
In general, there is no relationship.