If the equation `a x^2+b x+c=0` has two positive and real roots, then prove that the equation `a x^2+(b+6a)x+(c+3b)=0` has at least one positive real root.

If the equation ax^(2)+bx+c=0 has non-real roots,prove that 1+(c)/(a)+(b)/(a)>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.

Suppose a, b, c are real numbers, and each of the equations x^(2)+2ax+b^(2)=0 and x^(2)+2bx+c^(2)=0 has two distinct real roots. Then the equation x^(2)+2cx+a^(2)=0 has - (A) Two distinct positive real roots (B) Two equal roots (C) One positive and one negative root (D) No real roots

Prove that x^(6)+x^(4)+x^(2)+x+3=0 has no positive real roots.

If a, b, c are positive and a = 2b + 3c, then roots of the equation ax^(2) + bx + c = 0 are real for

If the equation ax^(2)+bx+c=x has no real roots,then the equation a(ax^(2)+bx+c)^(2)+b(ax^(2)+bx+c)+c=x will have a.four real roots b.no real root c.at least two least roots d.none of these

The equation |x-a|-b|=c has four distinct real roots,then

Prove that the equation,3x^(3)-6x^(2)+6x+sin x=0 has exactly one real root.

If a, b and c are positive real and a = 2b + 3c , then the equation ax^(2) + bx + c = 0 has real roots for