`x ^(2) +1` has no real zeroes. Why ?

The polynomial x^3 - kx^2 + 9x has three real zeros if and only if, k satifies

If ax^2 + bx + c = 0, a, b, c in R has no real zeros, and if a + b + c + lt 0, then

For a , b , c non-zero, real distinct, the equation, (a^(2)+b^(2))x^(2)-2b(a+c)x+b^(2)+c^(2)=0 has non-zero real roots. One of these roots is also the root of the equation :

A quadratic trinomial P(x)=a x^2+b x+c is such that the equation P(x)=x has no real roots. Prove that in this case equation P(P(x))=x has no real roots either.

If the equation a x^2+b x+c=x has no real roots, then the equation a(a x^2+b x+c)^2+b(a x^2+b x+c)+c=x will have a. four real roots b. no real root c. at least two least roots d. none of these

Let a ,b ,c be real. If a x^2+b x+c=0 has two real roots alphaa n dbeta,w h e r ealpha >1 , then show that 1+c/a+|b/a|<0

Prove that the equation x^(2)(a^(2)b^(2))+2x(ac+bd)+(c^(2)+d^(2))=0 has no real root if adnebc .

Late a in R and let f: Rvec be given by f(x)=x^5-5x+a , then f(x) has three real roots if a >4 f(x) has only one real roots if a >4 f(x) has three real roots if a<-4 f(x) has three real roots if -4

The equation alpha x^(3)-2(alpha+1) x^(2)+4 alpha x=0 has real roots and alpha is any positive integer, then the sum of the roots of the equation is