Algebra

Quadratic Discriminant

Quadratic Discriminant: Level 3 Challenges

         

f(x)=ax2+bx+cf(x)=ax^2+bx+c. a,b,ca,b,c are real numbers. Given that c(a+b+c)<0c(a+b+c)<0 , then what can we say about b24acb^2-4ac ?

Find the number of positive real roots of the equation x44x1=0.x^{4}-4x-1=0.

Given a function f(x)=ax2bx16f(x) = ax^{2}-bx-16 doesn't has 2 distinct real roots. Find the maximum value of 4ab4a-b.

k=1999(x247x+k)=(x247x+1)(x247x+2)(x247x+999)\prod_{k=1}^{999} (x^2-47x+k) = (x^2-47x+1)(x^2-47x+2)\dots(x^2-47x+999)

If the product of all real roots of the polynomial above can be expressed in the form n!n!, what is the value of nn?

Find the number of real roots of the polynomial 3x525x3+60x3x^{5} -25x^{3} +60x.

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