I need help with a radicals questions

Question

The roots of the equation $ax^2+bx+c=0$ are in the ratio of $2:3$. Determine an expression for $b$ in the terms of $a$, and $c$. I need help solving the question, please help, and thanks!

Answer 1

Let $x=u$ and $x=v$ be the solutions to the equation. Then you have $$ax^2+bx+c = a(x-u)(x-v) = ax^2-a(u+v)x + auv$$ If the roots are in the ratio of $2 : 3$, then we can assume that, say, $v=\frac{3u}{2}$. Substituting and comparing coefficients yields the answer to your question.

Answer 2

Alternately:

$$2(-b+\sqrt{b^2-4ac})=3(-b-\sqrt{b^2-4ac})$$

which simplifies to

$b = \pm \sqrt{\frac56ac}$