If `c` is positive and `2a x^2+3b x+5c=0` does not have any real roots, then prove that `2a-3b+5b>0.`

If c lt 0 and ax^2 + bx + c = 0 does not have any real roots then prove that: (i) a-b + c lt 0 , (ii) 9a +3b +c lt 0

If a, c gt 0 and ax^2 + 2bx + 3c = 0 does not have any real roots then prove that: (i) 4a-4b + 3c gt 0 , (ii) a + 6b + 27c gt 0 , (iii) a + 2b + 6c gt 0

If a x^2+b x+6=0 does not have distinct real roots, then find the least value of 3a+bdot

If a x^2+b x+6=0 does not have distinct real roots, then find the least value of 3a+bdot

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.

Given that a x^2+b x+c=0 has no real roots and a+b+c 0 d. c=0

Given that a x^2+b x+c=0 has no real roots and a+b+c 0 d. c=0

If 2x^3 +ax^2+ bx+ 4=0 (a and b are positive real numbers) has 3 real roots, then prove that a+ b ge 6 (2^(1/3)+ 4^(1/3))

a ,b ,c are positive real numbers forming a G.P. ILf a x^2+2b x+c=0a n ddx^2+2e x+f=0 have a common root, then prove that d//a ,e//b ,f//c are in A.P.

Column I, Column II If a ,b ,c ,a n dd are four zero real numbers such that (d+a-b)^2+(d+b-c)^2=0 and he root of the equation a(b-c)x^2+b(c-a)x=c(a-b)=0 are real and equal, then, p. a+b+c=0 If the roots the equation (a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0 are real and equal, then, q. a ,b ,ca r einAdotPdot If the equation a x^2+b x+c=0a n dx^3-3x^2+3x-0 have a common real root, then, r. a ,b ,ca r einGdotPdot Let a ,b ,c be positive real number such that the expression b x^2+(sqrt((a+b)^2+b^2))x+(a+c) is non-negative AAx in R , then, s. a ,b ,ca r einHdotPdot