## Relationship Between Coefficients and Root For Roots of a Quadratic in a Given Ratio

Suppose the roots of the quadratic
$x^2+ax+b$
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
$a=3x_2, \: b=2x_2^2$
.