Question on how to prove an expression satisfies a quadratic equation

Question

So I stumbled upon a question in a book that goes, "Prove that in any field, if $ax_1^2 + bx_1 + c = 0$ and $a ≠ 0$, then $x_2 = -(\frac {b}{a} + x_1)$ satisfies $ax_2^2 + bx_2 + c = 0$".

My question is, would merely substituting $x_2 = -(\frac{b}{a} + x_1)$ into the last equation be enough to complete the proof, or would I need to somehow derive the expression for $x_2$ from both the first and last equation?

Answer 1

As per michael-stachowsky's comment, it suffices to substitute the value for $x_2$ into the equation to complete the proof, but it may be simpler to note that, if $a(x-x_1)(x-x_2)=ax^2+bx+c,$ then from comparing coefficients $-a(x_1+x_2)=b$.

Answer 2

Alternatively: $$ax_1^2+bx_1+c=0=ax^2+bx+c \iff \\ a(x-x_1)(x-(-b/2-x_1))=0 \Rightarrow \\ x_1=x_1, x_2=-b/2-x_1.$$