If `tanthetaa n dsectheta` are the roots of `a x^2+b x+c=0,` then prove that `a^4=b^2(b^2-4ac)dot`

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

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

If cos (A /2) =sqrt((b+c)/(2c)) , then prove that a^2+b^2=c^2dot

If a ,b ,a n dc are in G.P. then prove that 1/(a^2-b^2)+1/(b^2)=1/(b^2-c^2)dot

If the roots of x^2-bx +c=0 are two consecutive integers then b^2-4c=

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 alpha,beta are the roots of the equation a x^2+b x+c=0, then find the roots of the equation a x^2-b x(x-1)+c(x-1)^2=0 in term of alphaa n dbetadot

If alpha,beta are the nonzero roots of a x^2+b x+c=0 and alpha^2,beta^2 are the roots of a^2x^2+b^2x+c^2=0 , then a ,b ,c are in (A) G.P. (B) H.P. (C) A.P. (D) none of these

If alpha is a real root of the quadratic equation a x^2+b x+c=0a n dbeta ils a real root of -a x^2+b x+c=0, then show that there is a root gamma of equation (a//2)x^2+b x+c=0 whilch lies between alpha & beta

If alpha and beta are the roots of the equation ax^(2)+bx+c=0 then the sum of the roots of the equation a^(2)x^(2)+(b^(2)-2ac)x+b^(2)-4ac=0 is