SOLUTION: {{{X^2-px+4>0}}} then |p|

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Question 1038912: X%5E2-px%2B4%3E0 then |p|
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
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X%5E2-px%2B4%3E0 then |p|
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  x%5E2+-px+%2B+4 = 

= %28x-%28p%2F2%29%29%5E2+-+%28p%2F2%29%5E2+%2B+4 = 

= %28x-%28p%2F2%29%29%5E2 + %284-%28p%2F2%29%5E2%29.   (1)

In (1), the addend  %28x-%28p%2F2%29%29%5E2  can be done an arbitrary small by choosing  "x",  but  the entire sum (1) remains positive, according to the condition.
It means that the second addend  %284-%28p%2F2%29%5E2%29 > 0,  or,  in other words,
%28p%2F2%29%5E2 < 4.

Hense,  abs%28p%2F2%29 < 2,   or  abs%28p%29 < 4.

QED.

Answer.  |p| < 4.

Three plots are shown below with p=3, 4, and 5.



Figure. Plots of functions y = x%5E2+-px+%2B+4, p = 3 (red), 4 (green) and 5 (blue).