\[x^2+ax+b\]
are in the ratio 2:1.What is the relationship between
\[a\]
and \[b\]
?Let the roots of the quadratic be
\[x_1, \: x_2\]
, then we can write the quadratic as \[(x-x_1)(x-x_2)\]
.Using the ratio condition, let
\[x_1=2x_2\]
so the last expression becomes \[(x-2x_2)(x-x_2)=x^2-2x_2-x_1+2x^2_2\]
and put this equal to the quadratic in question.\[x^2-2x_2-x_1+2x^2_2=x^2-3x_2+2x^2_2=x^2+ax+b\]
Then
\[a=3x_2, \: b=2x_2^2\]
.