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if c is positive and 2ax 2 +3bx+5c=0 does not have any real roots,then prove that 2a-3b+5c>0. Thanks!

if c is positive and 2ax2+3bx+5c=0 does not have any real roots,then prove that 2a-3b+5c>0.


Thanks!

Grade:11

2 Answers

Askiitians Expert Soumyajit IIT-Kharagpur
28 Points
11 years ago

 

Dear Palak J

Ans: discriminant part is >0 and hence (3b)^2 < 4*a*c and hence 3b>2(10a*c)^(0.5)

now 2a-3b+5c > 2a-2(10a*c)^(0.5)+5c

or 2a-3b+5c>(p+q)^2

where p=(2a)^(0.5)   & q=(5c)^(0.5)

As this is a square term and hence is>0

hence 2a-3b+5c>0(proved)

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Regards,

Askiitians Experts
Soumyajit Das IIT Kharagpur

mycroft holmes
272 Points
11 years ago

Since the given quadratic has no real roots, either f(x)>0 for all x or f(x)<0 for all x.

 

We are given that 5c = f(0)>0

 

Hence 2a-3b+5c = f(-1)>0

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