Question 1134883


Show that the equation {{{ (9k-8)x^2-6kx+k+2=0 }}} has roots which are different if {{{ k<8/5 }}}

discriminant {{{b^2-4ac }}}shows you when equation has {{{2}}}  different roots  

if you want distinct real roots, therefore the discriminant must be positive:

{{{b^2-4ac >0}}}

in your case {{{a=(9k-8)}}}, {{{b=6k}}}, and {{{c=k+2}}}

so, {{{(6k)^2-4(9k-8)(k+2) >0}}}

{{{36k^2-(36k-32)(k+2) >0}}}

{{{36k^2-(36 k^2 + 40 k - 64) >0}}}

{{{36k^2-36k^2 -40k +64 >0}}}

{{{ -40k +64 >0}}}

 {{{ 64 >40k}}}

{{{k<64/40}}}

{{{k<8/5}}}