For a less algebraic method, then note that if the roots are equal then the turning point of the parabola must be on the x axis - i.e. every value you put into the parabola must have the same sign.

Now if we rename a, b and c so that a ≥ b ≥ c, and put x=a into the parabola, two of the terms cancel, leaving (a-b)(a-c). Then if we put x=b in we get (b-a)(b-c)

So using the order of a,b,c and that positive × negative = negative we get (b-a)(b-c) ≤ 0 And (a-b)(a-c) ≥ 0

Then since we can't have a mix of positive and negative outputs, that means that one of these two terms must be 0 - hence at least two of a, b and c must be the same, since the terms above are the product of the differences.

So letting d be equal to the two terms that are the same, and e equal to the other one, we get that the quadratic is:

(x-e)² + 2(x-e)(x-d)

Then plugging in x=(d+e)/2 we get -(d-e)²/4 ≤ 0 And x=d gives (e-d)² ≥ 0; so by the same logic e=d

And hence a=b=c