If `P(x)=ax^(2)+bx+c` and `Q(x)=-ax^(2)+dx+c` where `ac!=0` then `P(x)Q(x)=0` has atleast

If P(x)=ax^2+bx+c , Q(x)=-ax^2+dx+c where ac!=0 then P(x).Q(x)=0 has

If P(x) = ax^(2) + bx + c , and Q(x) = -ax^(2) + dx + c, ax ne 0 then prove that P(x), Q(x) =0 has atleast two real roots.

Let a, b, c in R with a gt 0 such that the equation ax^(2) + bcx + b^(3) + c^(3) - 4abc = 0 has non-real roots. If P(x) = ax^(2) + bx + c and Q(x) = ax^(2) + cx + b , then (a) P(x) gt 0 for all x in R and Q(x) lt 0 for all x in R . (b) P(x) lt 0 for all x in R and Q(x) gt 0 for all x in R . (c) neither P(x) gt 0 for all x in R nor Q(x) gt 0 for all x in R . (d) exactly one of P(x) or Q(x) is positive for all real x.

Statement-1: If alpha and beta are real roots of the quadratic equations ax^(2) + bx + c = 0 and -ax^(2) + bx + c = 0 , then (a)/(2) x^(2) + bx + c = 0 has a real root between alpha and beta Statement-2: If f(x) is a real polynomial and x_(1), x_(2) in R such that f(x_(1)) f_(x_(2)) lt 0 , then f(x) = 0 has at leat one real root between x_(1) and x_(2) .

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

If p(x)=ax^(2)+bx and q(x)=lx^(2)+mx+n with p(1)=q(1), p(2)-q(2)=1 , and p(3)-q(3)=4 , then p(4)-q(4) is equal to

Let f(x)=ax^(2)+bx+c , g(x)=ax^(2)+px+q , where a , b , c , q , p in R and b ne p . If their discriminants are equal and f(x)=g(x) has a root alpha , then

If f(x) = a + bx + cx^(2) where c > 0 and b^(2) - 4ac < 0 . Then the area enclosed by the coordinate axes,the line x = 2 and the curve y = f(x) is given by

Consider two quadratic expressions f(x) =ax^2+ bx + c and g (x)=ax^2+px+c,( a, b, c, p,q in R, b != p) such that their discriminants are equal. If f(x)= g(x) has a root x = alpha , then

Given P(x) =x^(4) +ax^(3) +bx^(2) +cx +d such that x=0 is the only real root of P'(x) =0 . If P(-1) lt P(1), then in the interval [-1,1]