Wrong Solution

Source: Very interesting problem taken from Tanya Khovanova’s Math Blog

Problem:
I found this cute problem in the Russian book Sharygin Geometry Olympiad by Zaslavsky, Protasov and Sharygin.
Find numbers p and q that satisfy the equation: x2 + px + q = 0.
The book asks you to find a mistake in the following solution:
By Viète’s formulae we get a system of equations p + q = − p, pq = q. Solving the system we get two solutions: p = q = 0 and p = 1, q = −2.
What is wrong with this solution?

Update (26-05-2011)
Solution:
Posted by Sudeep Kamath (EECS PhD Student, UC Berkeley, EE IITB 2008 Alumnus) in comments and of course on Tanya Khovanova’s Math Blog


Comments

  1. shaan8:37 PM, May 23, 2011

    here, we are looking for the coefficients "p" & "q" most probably,
    i think the question DOES NOT that
    P & q ARE THE SOLUTIONS OF GIVEN QUADRATIC..

    ReplyDelete
  2. Sudeep10:37 PM, May 23, 2011

    Very nice problem.

    This approach misses out the case when p=q, so that you only need to enforce one root of the equation to be p. This gives the additional solution p=q=-1/2.

    ReplyDelete
  3. Pratik Poddar2:46 PM, May 26, 2011

    @Shaan. I did not understand your comment.

    @Sudeep. \m/ \m/ I could not solve it, but was happy_max when I read the solution :) Thanks.

    ReplyDelete
  4. satpal10:48 AM, May 11, 2012

    but when you put p=q=-1/2 in the equation the roots of the quadratic are coming out to be -1/2 and 1 which are distinct and we need the equal roots..

    ReplyDelete
  5. Pratik Poddar4:55 AM, May 25, 2012

    @satpal..
    When you put p = q = -1/2,
    The equation becomes 2x^2 - x - 1 = 0 and -1/2 is a root of this equation. The question never says that p and q are the only roots of the equation. :)

    Its just a terminology difference. I would not have put this if this was not in Russian Olympiad book :)

    ReplyDelete

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