Determine whether there exist distinct real numbers
![a, b, c, t](/media/m/9/b/3/9b37f63dec4850ef5b88f4d9abbee934.png)
for which:
(i) the equation
![ax^2 + btx + c = 0](/media/m/f/1/1/f11a7f5fa9a0ea2bcda1152bacdb4762.png)
has two distinct real roots
![x_1, x_2,](/media/m/3/0/0/3004052e3b02d1095b01ea7866e1157f.png)
(ii) the equation
![bx^2 + ctx + a = 0](/media/m/2/d/6/2d6a4aebd5a72b86f5d9969928ff83ad.png)
has two distinct real roots
![x_2, x_3,](/media/m/0/d/2/0d2daf9a464bb5e8195c5a0a7bb51985.png)
(iii) the equation
![cx^2 + atx + b = 0](/media/m/4/1/4/4149a3437013a6c4e5db089e070befe3.png)
has two distinct real roots
%V0
Determine whether there exist distinct real numbers $a, b, c, t$ for which:
(i) the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$
(ii) the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$
(iii) the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$