What is the role played by the difference

Karen Smith

Karen Smith

Answered question

2022-03-12

What is the role played by the difference of the two roots in solving a quadratic equation?
I have to write an essay and this is the main question of the essay, and I'm massively stuck.
All I have so far is the difference of squares theorem, which I think might just be massively irrelevant. The module is Using Radicals to Solve and I not only don't understand the question, I can't find any relevant resources.
I just don't think I'm going in the right direction so it would be so so so appreciated if somebody could give me their interpretation of what I'm being asked, or even point me in the right direction of some online resources etc.

Answer & Explanation

klepbroek31s

klepbroek31s

Beginner2022-03-13Added 4 answers

If x1,x2 are the two roots of a quadratic polynomial
X2+pX+q,
then their difference is
δ=x1x2,
of course. The essay could end here, but note that the roots do not form an ordered pair, i.e., we might as well swap their order, which produces a different δ, namely x2x1. This is almost the same, namely minus the original δ. What do we do to get rid of the uncertainty of the sign? We square! So the more natural expression in terms of difference of roots is D=δ2=(x1x2)2. This D does not depend on the order of x1,x2; we say that it is symmetric in x1,x2. As it expands as D=x12+x222x1x2, this is clear anyway.

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