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

True or false The equation 2x^2+3x+1=0 has an irrational root. If a

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+a^2a b a c a b x+b^2b c a c b c x+c^2|,t h e n prove that f^(prime)(x)=3x^2+2x(a^2+b^2+c^2)dot

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)=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 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 .

Two distinct polynomials f(x) and g(x) defined as defined as follow : f(x) =x^(2) +ax+2,g(x) =x^(2) +2x+a if the equations f(x) =0 and g(x) =0 have a common root then the sum of roots of the equation f(x) +g(x) =0 is -

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 f(x)=x^(2)+x+(3)/(4) and g(x)=x^(2)+ax+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)