IIT JEE , AIEEE Forum - Mathematics , Algebra

sankydreams

Find out minimum non-negative real values of a,b and c given that the eqn x⁴+ ax³ + bx² +cx + 1 = 0 has real roots.

sankydreams

hey experts, where is the solutions..................

puneet

hii

I make some observations like .... for x > 0 it will be always greater than 0.

So, roots must be negative.

Also since product of these roots is one atleast one root must lie in 0 to -1 range ..

So, now u try .. i am also trying ..

cheers

raja987654321

Let x⁴+ax³+bx²+cx+1=(x²+mx+1)(x²+nx+1)=0..........(i)

Clearly , a= m+n ; c= m+n ; b=2......(ii)

Since the mother equation above has only real roots so has her sons

Hence m²>=4 ; n²>=4 ........(iii)

Again a,b,c>0 i.e. the least positive values of a,b,c satifying the given conditions is

a=4 ;b=2;c=4 ..........[Since m=2,n=2 from (i),(ii)&(iii) ]