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?

  • $\begingroup$ This page has a post that will tell you everything you need to know about quadratic equations: math.stackexchange.com/questions/49229/… $\endgroup$ – астон вілла олоф мэллбэрг Apr 6 '16 at 1:11
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    $\begingroup$ That’s a great observation to make without being taught it. The quadratic formula gives you the values $x_1$ and $x_2$. What happens if you write the $+$ and $-$ roots out from the formula and then add them to see what the quadratic formula tells you $x_1+x_2$ is? (It is known, but it’s still satisfying to find these things on your own.) $\endgroup$ – Steve Kass Apr 6 '16 at 1:23
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    $\begingroup$ Hint: if the roots are $\{x_1, x_2\}$ then you can write: $(x - x_1)(x - x_2) = 0$--what is $b$ in this case?? $\endgroup$ – Jared Apr 6 '16 at 1:32
  • $\begingroup$ It happens again and again: what we once thought to be “weird” turns out to be fundamental. $\endgroup$ – Lubin Apr 6 '16 at 1:36

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 $a=1$.


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