Arithmetic sequence with polynomial

Question

I need assistance with the following question:

Let $a,b,c$ be $3$ consecutive terms of an arithmetic progression. We know that one root of $ax^2+bx+c=0$ is $-5$ what is the second root?

I tried to write the terms like this: $a$, $a+d$, $a+2d$ and I plugged it to quadratic equation.

Answer 1

Plug $x=-5$ to find: $$25a-5b+c=0$$ Since $a,b,c$ form arithmetic progression, then: $$a+c=2b$$ From these two we can get: $$c=15a$$ Now the second root is by the Vieta's formula: $$-5x_2=\frac{c}{a} \Rightarrow x_2=-\frac15\cdot \frac{15a}{a}=-3.$$

Answer 2

Hint. Let $r$ be the second root then $$\begin{cases}b/a=-(-5+r)\\ c/a=-5\cdot r\end{cases}$$ Now let $b=a+d$ and $c=a+2d$ and solve the linear system with respect to the unknowns $r$ and $d/a$. What is the value of $r$?