If `a x^2+(b-c)x+a-b-c=0` has unequal real roots for all `c in R ,t h e n`

If x^2+(a-b)x+(1-a-b)=0. w h e r ea ,b in R , then find the values of a for which equation has unequal real roots for all values of bdot

If ax^(2) + bx + c = 0 , a , b, c in R has no real roots, and if c lt 0, the which of the following is ture ? (a) a lt 0 (b) a + b + c gt 0 (c) a +b +c lt 0

Let a in R and f : R rarr R be given by f(x)=x^(5)-5x+a , then (a) f(x)=0 has three real roots if a gt 4 (b) f(x)=0 has only one real root if a gt 4 (c) f(x)=0 has three real roots if a lt -4 (d) f(x)=0 has three real roots if -4 lt a lt 4

Let a ,b ,c be real. If a x^2+b x+c=0 has two real roots alphaa n dbeta,w h e r ealpha >1 , then show that 1+c/a+|b/a|<0

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 x^2+3x+5=0a n da x^2+b x+c=0 have common root/roots and a ,b ,c in N , then find the minimum value of a+b+c

If ax^2 + bx + c = 0, a, b, c in R has no real zeros, and if a + b + c + lt 0, then

If a ,b ,c ,d in R , then the equation (x^2+a x-3b)(x^2-c x+b)(x^2-dx+2b)=0 has a. 6 real roots b. at least 2 real roots c. 4 real roots d. none of these

If alpha,beta are real and distinct roots of a x^2+b x-c=0a n dp ,q are real and distinct roots of a x^2+b x-|c|=0,w h e r e(a >0), then alpha,beta in (p ,q) b. alpha,beta in [p ,q] c. p ,q in (alpha,beta) d. none of these

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.