If `f(x)=a x^2+b x+c ,g(x)=-a x^2+b x+c` ,where ac`!=0,` then prove that `f(x)g(x)=0` has at least two real roots.

If b^2<2a c , then prove that a x^3+b x^2+c x+d=0 has exactly one real root.

If f(x)=x^2+x+3/4 and g(x)=x^2+a x+1 be two real functions, then the range of a for which g(f(x))=0 has no real solution is (A) (-oo,-2) (B) (-2,2) (C) (-2,oo) (D) (2,oo)

Let f (x) =ax ^(2) +bx + c,a ne 0, such the f (-1-x)=f (-1+ x) AA x in R. Also given that f (x) =0 has no real roots and 4a + b gt 0. Let p =b-4a, q=2a +b, then pq is:

Let f(x) = x2 + b_1x + c_1. g(x) = x^2 + b_2x + c_2 . Real roots of f(x) = 0 be alpha, beta and real roots of g(x) = 0 be alpha+gamma, beta+gamma . Least values of f(x) be - 1/4 Least value of g(x) occurs at x=7/2

Let f(x) = x2 + b_1x + c_1. g(x) = x^2 + b_2x + c_2 . Real roots of f(x) = 0 be alpha, beta and real roots of g(x) = 0 be alpha+gamma, beta+gamma . Least values of f(x) be - 1/4 Least value of g(x) occurs at x=7/2

If f(x)=x^3+b x^2+c x+d and f(0),f(-1) are odd integers, prove that f(x)=0 cannot have all integral roots.

Let f(x)=x^(2)+bx+c and g(x)=x^(2)+b_(1)x+c_(1) Let the real roots of f(x)=0 be alpha, beta and real roots of g(x)=0 be alpha +k, beta+k fro same constant k . The least value fo f(x) is -1/4 and least value of g(x) occurs at x=7/2 The roots of g(x)=0 are

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

If a+b+c=0(a,b, c are real), then prove that equation (b-x)^2-4(a-x)(c-x)=0 has real roots and the roots will not be equal unless a=b=c .

Let f(x)=ax^(2)+bx+c , g(x)=ax^(2)+qx+r , where a , b , c , q , r in R and a lt 0 . If alpha , beta are the roots of f(x)=0 and alpha+delta , beta+delta are the roots of g(x)=0 , then