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

Suppose the roots of the quadratic  
  are in the ratio 2:1.
What is the relationship between  
Let the roots of the quadratic be  
\[x_1, \: x_2\]
, then we can write the quadratic as  
Using the ratio condition, let  
  so the last expression becomes  
  and put this equal to the quadratic in question.

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