20th VMO 1982

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Problem A2

Find all real solutions to x(x + 1)(x + 2)(x + 3) = m - 1.

 

Answer

m < 0, no solutions
m = 0, two solutions, x = -3±√5)/2
0 < m < 25/16, four solutions, x = -3/2 ±√(5/4 ±√m)
m = 25/16, three solutions, x = -3/2, -3/2 ±√(5/2)
m > 25/16, two solutions, x = -3/2 ±√(5/4 + √m)

 

Solution

We have m = (x2+3x+1)2 = (x + 3/2)2 - 5/4. So if m is negative there are no solutions. For non-negative m we have solutions as above

Thanks to Suat Namli

 


 

19th VMO 1981

© John Scholes
jscholes@kalva.demon.co.uk
15 February 2004
Last corrected/updated 15 Feb 04