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

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

If the quadratic equation a x^2+b x+6=0 does not have real roots and b in R^+ , then prove that a > m a x{(b^2)/(24),b-6}

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))

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 b^2<2a c , then prove that a x^3+b x^2+c x+d=0 has exactly one real root.

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.

a ,b ,c are positive real numbers forming a G.P. ILf a x 62+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.

If the equation a x^2+b x+c=0,a ,b ,c , in R have none-real roots, then a. c(a-b+c)>0 b. c(a+b+c)>0 c. c(4a-2b+c)>0 d. none of these

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

If a_1x^3 + b_1x² + c_1x + d_1 = 0 and a_2x^3 + b_2x^2+ c_2x + d_2 = 0 have a pair of repeated roots common, then prove that |[3a_1,2b_1,c_1],[3a_2,2b_2,c_2],[a_2b_1-a_1b_2,c_1a_2-c_2a_1,d_1a_2-d_2a_1]|=0