If the roots of `(a-b)x^(2)+(b-c)x+(c-a)=0` are real and equal, then prove that b, a, c are in arithmetic progression.

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

If the roots of the equation (c^(2)-ab)x^(3)-2(a^(2)-bc)x+b^(2)-ac=0 are real and equal prove that either a=0 (or) a^(3)+b^(3)+c^(3)=3"abc" .

If a,b,c are in geometric progression and if a^((1)/(x))=b^((1)/(y))=c^((1)/(z)) , then prove that x,y,z are in arithmetic progression.

If a, b, c are in geometric progression, and if a^(1/x)=b^(1/y)=c^(1/z) , then prove that x, y, z are in arithmetic progression.

If the roots of the equation a(b-c)x^2+b(c-a)x+c(a-b)=0 are equal, show that 2//b=1//a+1//c dot

If a,b,c are in geometric progression, and if a^(1/x) = b^(1/y) =c^(1/z) , then prove that x,y,z arithmetic progression.

If two roots of x^3-a x^2+b x-c=0 are equal inn magnitude but opposite in signs, then prove that a b=c

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 x is real and the roots of the equation a x^2+b x+c=0 are imaginary, then prove tat a^2x^2+a b x+a c is always positive.

If s intheta,costheta be the roots of a x^2+b x+c=0 , then prove that b^2=a^2+2ac.