If n and r are positive of the equation `x^2-bx+c=0` then show that the roots of the quadratic equatin`^cC_r x^2+2. ^nC_(r+1)x+^nC_(r+2)=0` are real.

If n and r are positive integers such that 0ltrltn then show that the roots of the quadratic equation nC_r x^2+2. ^nC_(r+1)x+^nC_(r+2)=0 are real.

Show that the roots of the equation x^3 +px^2 +qx +r=0 are in H.P then 2q^3 =9r (pq-3r)

If p, q, r each are positive rational number such tlaht p gt q gt r and the quadratic equation (p + q - 2r)x^(2) + (q + r- 2p)x + (r + p - 2q) = 0 has a root in (-1 , 0) then which of the following statement hold good? (A) (r + p)/(q) lt 2 (B) Both roots of given quadratic are rational (C) The equation px^(2) + 2qx + r = 0 has real and distinct roots (D) The equation px^(2) + 2qx + r = 0 has no real roots

if alpha&beta are the roots of the quadratic equation ax^2 + bx + c = 0 , then the quadratic equation ax^2-bx(x-1)+c(x-1)^2 =0 has roots

If a, b, c in R and the quadratic equation x^2 + (a + b) x + c = 0 has no real roots then

If r is the ratio of the roots of the equation ax^2 + bx + c = 0 , show that (r+1)^2/r =(b^2)/(ac)

If a,b,c in R and the equation ax^2+bx+c = 0 and x^2+ x+ 1=0 have a common root, then

If p , q , ra n ds are real numbers such that p r=2(q+s), then show that at least one of the equations x^2+p x+q=0 and x^2+r x+s=0 has real roots.

If alpha and beta are the roots of the quadratic equation 4x ^(2) + 2x -1=0 then the value of sum _(r =1) ^(oo) (a ^(r ) + beta ^(r )) is :

Minimum possible number of positive root of the quadratic equation x^2 -(1 +lambda)x+ lambda -2 =0 , lambda in R