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Discriminant with inequality proof


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#1 O.J

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Posted 8 June 2016 - 10:06 AM

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, 8 June 2016 - 03:38 PM.

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#2 mathematic

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Posted 8 June 2016 - 11:01 PM

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


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#3 Keen

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Posted 28 June 2016 - 12:43 PM

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, 28 June 2016 - 12:46 PM.

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#4 blue89

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Posted 4 September 2016 - 12:54 PM

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 :

 

x+ 4  = ax+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  

 

x= 2i   x2 = - 2i    both x& x2  ϵ ₵

 

and how did you compare with zero ?

 

( when you say x1  ,xare >0 ??) 


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