Question

# Show that if roots of equation ( left(a^{2}-b cright) x^{2}+2left(b^{2}-a cright) x+c^{2}-a b=0 ) are equal, then either ( b=0 ) or ( a^{3}+b^{3}+c^{3}=3 a b c )

Solution

( D=0 )

( left(2left(b^{2}-a cright)right)^{2}-4left(a^{2}-b cright)left(c^{2}-9 bright)=0 )

( 4left(left[b^{2}-9 cright)^{2}-left(a^{2}-6 cright)left(c^{2}-a bright)right]=0 )

( 6^{4}+9^{2} c^{2}-29 b^{2} c-left(9^{2} c^{2}-6 c^{3}-9^{3} b+9 b^{2} c^{2}right) )

( =0 )

( 6^{4}+9^{2} k^{2}-29 b^{2} c-9^{2} c^{2}+6 c^{3}+9^{3} b^{2} c= )

( 6left(6^{3}-396 c+9^{3}+c^{3}right)=0 )

then ( 6=0 ) of ( a^{3}+b^{3}+c^{3}-3 a b c )