Birthday problem
Algebra
Level
4
If
has all real roots then the maximum value of c is
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2
4
5
1
6
8
3
7
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1 solution
Let x = y − 1 . Then x 4 + 4 x 3 + 2 x 2 c + 4 x c − 8 x + c 2 − 4 c + 4 = ( y − 1 ) 4 + 4 ( y − 1 ) 3 + 2 ( y − 1 ) 2 c + 4 ( y − 1 ) c − 8 ( y − 1 ) + ( c − 2 ) 2 = y 4 − 4 y 3 + 6 y 2 − 4 y + 1 + 4 y 3 − 1 2 y 2 + 1 2 y − 4 + 2 c y 2 − 4 c y + 2 c + 4 c y − 4 c − 8 y + 8 + ( c − 2 ) 2 = y 4 + 2 ( c − 3 ) y 2 + c 2 − 6 c + 9 = ( y 2 + c − 3 ) 2 Since y must be real, y 2 = 3 − c ≥ 0 ⇒ c ≤ 3