Today, we learnt about the quadratic formula, and I noticed a strange property of (seemingly) all quadratic equations. If we call the two solutions of any arbitrary quadratic equation $x_1$ and $x_2$:
$$When \ a=1: b=-(x_1+x_2)$$ Why is this? I've tried some feeble attempts at solving this algebraically but nothing has come close to working. I can guess that the answer (as is the case with a lot of these odd, seemingly coincidental identities) will lie in a geometric interpretation of the quadratic formula. No research has yielded any results.
So my questions are: Is this a known result? And how is this identity derived or explained?
All quadratic equations are in the form of $$a(x-x_1)(x-x_2).$$ In the special case of $a=1$ the equation is simply $$(x-x_1)(x-x_2).$$ Expanding gives $$x^2-x_1x-x_2x+x_1x_2,$$ and combining like-terms gives $$x^2-(x_1+x_2)x+x_1x_2.$$ So $b=-(x_1+x_2)$ and $c=x_1x_2$ when $a=1$.