If `a ,b ,c in R` such that `a+b+c=0a n da!=c` , then prove that the roots of `(b+c-a)x^2+(c+a-b)x+(a+b-c)=0` are real and distinct.

If a

If the roots of (a-b)x^(2)+(b-c)x+(c-a)=0 are equal, prove that 2a=b+c .

If a ,b ,c in R^+a n d2b=a+c , then check the nature of roots of equation a x^2+2b x+c=0.

The equation a x^2+b x+c=0 has real and positive roots. Prove that the roots of the equation a^2x^2+a(3b-2c)x+(2b-c)(b-c)+a c=0 re real and positive.

If a ,b ,c are three distinct real numbers in G.P. and a+b+c=x b , then prove that either x >3.

If a ,b ,c are three distinct real numbers in G.P. and a+b+c=x b , then prove that either x >3.

If roots of equation x^2+2c x+a b=0 are real and unequal, then prove that the roots of x^2-2(a+b)x+a^2+b^2+2c^2=0 will be imaginary.

If a , b , c are positive numbers such that a gt b gt c and the equation (a+b-2c)x^(2)+(b+c-2a)x+(c+a-2b)=0 has a root in the interval (-1,0) , then

If aa n dc are the lengths of segments of any focal chord of the parabola y^2=2b x ,(b >0), then the roots of the equation a x^2+b x+c=0 are (a)real and distinct (b) real and equal (c)imaginary (d) none of these

If a ,b ,c in R , then find the number of real roots of the equation =|x c-b-c x a b-a x|=0