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P&B #155

Let a,b,c be real numbers. Prove that three roots of the equation
b+cxa+c+axb+a+bxc=3
are real.

Solution: The solution which the book lists is actually much more elegant than what I will present here. We let s=a+b+c to be the sum. Inserting this into the equation, we see that
saxa+sbxb+scxc=3
Now, multiplying both sides by (xa)(xb)(xc) and rearranging results in
(sa)(xb)(xc)+(sb)(xa)(xc)+(sc)(xa)(xb)3(xa)(xb)(xc)=0.
By inspection, it’s pretty clear that x=s is a root, hence if we find one additional real root we are done as imaginary roots must exist in pairs.

Here is where I deviated against the solutions. I plugged in x=a,b,c respectively and considered all the cases separated by magnitude and showed that there must exist a pair to the left/right of s such that it is positive and negative, hence there exists another real root.

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