O.J 0 Posted June 8, 2016 (edited) 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 June 8, 2016 by O.J Share this post Link to post Share on other sites

mathematic 65 Posted June 8, 2016 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 Posted June 28, 2016 (edited) 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 June 28, 2016 by Keen Share this post Link to post Share on other sites

blue89 19 Posted September 4, 2016 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^{2 }+ 4 = ax^{2 }+bx +c , there a = 1 ( > 0 ) ( b= 0 no problem) and c= 4 , ∆ < 0 ( ∆ = - 16) (as you would) then look at the solution x_{1}= (-b +√ ∆)/2 x_{2} = (-b -√ ∆)/2 x_{1 }= 2i x_{2} = - 2i both x_{1 }& x_{2 }ϵ ₵ and how did you compare with zero ? ( when you say x_{1 },x_{2 }are >0 ??) Share this post Link to post Share on other sites