If the roots of the quadratic equation (4p-p^2 -5)x^2

Gage Potter

Gage Potter

Answered question

2022-04-24

If the roots of the quadratic equation
(4pp25)x2(2p1)x+3p=0
lie on either side of unity, then the number of integral values of p is?

Answer & Explanation

Aiyana Richmond

Aiyana Richmond

Beginner2022-04-25Added 14 answers

First notice that leading coefficient
a=(p24p+4+1)=(p2)21 is negative.
If the roots of quadratic function
f(x)=(4pp25)x2(2p1)x+3p
are x1 and x2 (say x1>x2) then we have
x2<1<x11x2>0 and 1x1<0
So if we write write f in factor form:
f(x)=a(xx1)(xx2)
then we see that
f(1)=a(1x1)(1x2)>0

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