The must be a faster way - 1

If p,q,rp,q,r are the roots of the equation


Prove that p=q=rp=q=r.

Elementary proof:

By Vieta's Formula,




From the third equation,


Substitute to the second equation,





Substitute pr=q2pr=q^2 into the third equation,

q3=r3    q=rq^3=r^3\implies q=r




p=r    p=q=rp=r \implies p=q=r

Can you find a better proof?

Note by Christopher Boo
5 years ago

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1 vote

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Put x=rx=r in the equation. Since rr is a root, we immediately get pr=q2pr=q^2 (assuming r0r\neq 0). Also pqr=r3pqr=r^3. The rest follows from these two.

Abhishek Sinha - 5 years ago

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Brilliant proof!!!

Anuj Shikarkhane - 5 years ago

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Brilliant! @Abhishek Sinha

Christopher Boo - 5 years ago

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we can assume that p=q . Which means that we are assuming p to be the repeated root of the given function . Differentiate the given cubic and let it be g(x). then substitute p in g(x) Since we have assumed p to be repeated root thus it will also be the root of g(x). on substituting p in the equation we will get p=q which concurs with our assumption.

Prince Kumar Maurya - 4 years, 9 months ago

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Indeed, I've found a much faster solution. Using the cubic formula, we get that


is one of pp, qq, and rr. We can find the other roots by dividing out, and the rest of the proof is omitted.

Cody Johnson - 5 years ago

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Honestly, why do you love to bash out all the problems so much?

Sagnik Saha - 5 years ago

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