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

\[a=3x_2, \: b=2x_2^2\]