There can't be only one, can there?

Algebra Level 4

Determine $m$ given that

$(x^2-2mx-4(m^2+1))(x^2-4x-2m(m^2+1))=0$

has exactly three different integral roots.

The answer is 3.

1 solution

Anish Mahendru
May 19, 2014

two quadratic equations share a common integral root. moreover roots of first quadratic is of the form m+-srt(5m**2+4) ,which makes m an integral

m=-1 est une solution

Omar El Mokhtar - 7 years ago

$m=\,-1$ implies the equation becomes $(x+4)(x-2)^3$ which clearly has $2$ instead of $3$ roots.

Nishant Sharma - 7 years ago

I think -1 is a solution, too

Younes MouMen - 6 years, 10 months ago

-1 could have been a solution if the system had exactly 2 different integral solutions. Initially the approach appears to be correct based on the assumption that one of the quadratic has equal roots. Logically, the second quadratic is the only one capable of having its discriminant=0. But after solving we find that the equal roots are in fact equal to one of the roots of the first quadratic.

Chandan Mahapatra - 6 years, 10 months ago

There can't be only one, can there?

The answer is 3.

1 solution

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