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I am writing a proof for the Theorem (x-a)(x+a)=x^2-a^2 and directly proved it by manipulating the equation using multiplication and addition axioms. But I'm not sure what should be included in the intro and conclusion.

user103748
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  • What do you mean by writing a proof professionally ? – ILoveMath Oct 28 '13 at 07:59
  • i am supposed to write a professional flowing proof with an intro, theorem stated, theorem proved, and then concluded – user103748 Oct 28 '13 at 08:03
  • I think all you should do is at each step simply write which axiom you're using. I don't think there is a need for an intro and conclusion. At the end just draw a little empty square - that usually signals that you've proved what you needed to prove. –  Oct 28 '13 at 08:26

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By two applications of distributivity, $(x-a)(x+a) = x(x+a) - a(x + a) = x^2 + xa - ax - a^2 $.

Since $\mathbb{R}$ (i assume you are playing around in the reals) is abelian, then $ab = ba $ holds for any $a,b$. In particular $ab - ba = 0 $.

Apply this to first paragraph to obtain $x^2 ( xa - ax) - a^2 = x^2 + 0 - a^2 = x^2 - a^2 $

Profesionally, then the problem has been solved.

ILoveMath
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  • i've already proved the theorem, I'm just wondering were i could get some good examples similar to this theorem. I'm not sure what the intro and conclusion should have in it – user103748 Oct 28 '13 at 08:13