RMO 2015! #13

Algebra Level 4

Find the sum of all positive integers n n for which the equation n x 4 + 4 x + 3 = 0 nx^4 + 4x + 3 = 0 has a real root.


The answer is 1.

Dev Sharma by Dev Sharma , 20, India

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2 solutions

Kush Singhal
Nov 28, 2015

For the solutions to be real, the discriminant has to be greater than or equal to 0. => 4 2 − 12 n > = 0 4^{2} - 12n >= 0 => 16 > = 12 n 16 >= 12n => 4 3 > = n > 0 \frac{4}{3} >= n > 0 as n is positive. Therefore n = 1

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the problem is x^4 not x^2

Eric Louis - 3 years ago
Achal Jain
Jun 21, 2016

I believe the best way is to put x 4 = m 2 x^{4} =m^{2} fro some integer m. Hence the new equation would be

n m 2 + 4 m + 3 { nm }^{ 2 }+4\sqrt { m } +3 . The just apply the formula to find the root as − 4 ± 16 − 12 n 2 n \frac { -4\pm \sqrt { 16-12n } }{ 2n } .

Hence only possible value is n = 1 n=1

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