O.J

Discriminant with inequality proof

Recommended Posts

O.J    0

Hi

 

I'm trying to solve this :

assume:  a*x^2+b*x+c>=0 for all x with a≠0 then
have: b^2-4*a*c<=0


but I couldn't. I proved it when the quadratic equation is greater than zero

a*x^2+b*x+c>0 for all x with a≠0

but not for greater then or equal to zero. So any one help in solving this please.

Edited by O.J

Share this post


Link to post
Share on other sites
mathematic    66

If the quadratic = 0 for some x and is never < 0, then it has a double root at that point and the discriminant=0.

Share this post


Link to post
Share on other sites
Keen    6

This is typically a statement that is best proven by contraposition. Assume that the discriminant is positive and then prove that your polynomial cannot be non negative in all points.

 

When you have a positive discriminant, your polynomial has exactly two roots and depending on the sign of a, it is either positive between these roots or negative. If it is negative, you're done. If not then it must be positive between the largest of the two roots and infinity and between minus infinity and the smallest of your two roots.

 

I guess you could also prove it directly, but then you'd probably have to waste time with who knows how many different cases.

 

Hope it is clear enough.

Edited by Keen

Share this post


Link to post
Share on other sites
blue89    19

Hi

 

I'm trying to solve this :

assume:  a*x^2+b*x+c>=0 for all x with a≠0 then
have: b^2-4*a*c<=0


but I couldn't. I proved it when the quadratic equation is greater than zero

a*x^2+b*x+c>0 for all x with a≠0

but not for greater then or equal to zero. So any one help in solving this please.

 

 

you seem like you are missing something or we are unsure about something.

 

look please this example :

 

x2 + 4 = ax2 +bx +c , there a = 1 ( > 0 ) ( b= 0 no problem) and c= 4 , ∆ < 0 ( ∆ = - 16) (as you would) then look at the solution

 

x1= (-b +√ ∆)/2 x2 = (-b -√ ∆)/2

 

x1 = 2i x2 = - 2i both x1 & x2 ϵ ₵

 

and how did you compare with zero ?

 

( when you say x1 ,x2 are >0 ??)

Share this post


Link to post
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now