Complex Roots $\alpha$ and $\beta$ satisfy the equation $(x-\alpha)(x-\beta) = 0$ but not $(x+\alpha)(x+\beta) = 0$

Question

I would like to ask a question about Complex Roots in Further Mathematics.

I am new to the subject and one of the statements given in the book without further explanation is

If the roots of the equation are $\alpha$ and $\beta$, the equation is $(x-\alpha)(x-\beta) = 0$

Hence, $(x-\alpha)(x-\beta) = x^2 – \alpha x – \beta x + \alpha \beta = x^2 – (\alpha + \beta)x + \alpha \beta$

Now, I have a question.
Why is the equation given as:

$$(x-\alpha)(x-\beta) = 0$$

and not as

$$(x+\alpha)(x+\beta) = 0$$

Is there a reason for this?

Answer 1

Inserting $\alpha$ into $(x-\alpha)(x-\beta)$ gives you $(\alpha-\alpha)(\alpha-\beta)$, which you can quickly confirm is equal to $0$. Same with inserting $\beta$.

On the other hand, inserting into $(x+\alpha)(x+\beta)$ doesn't give you zero (except in very special circumstances). So in order to make sure that $\alpha$ and $\beta$ are indeed roots, $-$ is the operation of choice.