Use quantifiers and logical connectives to express the fact that a quadratic polynomial with real number coefficients has at most two real roots.

There is really no need for derivatives. Here is the  

∀a,b,c ∈ ℝ, 
[[∃x,y,z ∈ ℝ, ax^2 bx c=ay^2 by c=az^2 bz c=0] ⇒ [(a-b)(b-c)(c-a)=0]]. 

Note: "(a-b)(b-c)(c-a)=0" is simply the same to saying "[a=b] ∨ [b=c] ∨ [c=a]", but a few keystrokes shorter.

you have to email it to yourself from your phone then download it to either google drive or you computer files. then you can go to where ever you saved it and select it and then copy and paste it. or you could attach it to your question when you are writing it which would be much easier. for example: go to the attachment.

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