How do pupils solve 2nd degree equations in Germany? (different from Spain) I'm from Spain and in Spain the undergraduate pupils learn to solve a 2nd degree (i.e. quadratic) equation using the formula 
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ 
but years ago I had a colleague who did secondary school in Germany and he solved this kind of equation using another formula/method.
Someone could tell me about the method taught to German pupils to solve 2nd degree equations? Using formulas, not graphical solutions.
Thank you.
 A: The other two common methods for solving a second degree equation are:


*

*Completing the square (which is essentially equivalent to using the aforementioned formula: this is basically how the formula is derived):


$$ax^2+bx+c=0$$
$$x^2+2\frac{b}{2a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}$$
$$(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}$$
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$


*

*If $x_1,x_2$ are the two solutions of the equation, it is known that


$$x_1+x_2=-\frac{b}{a}$$
$$x_1\cdot x_2=\frac{c}{a}$$
     This system of equation is easily solved by substitution or by any other mean.
I believe that your German friend might be referring to the latter method. Pupils are usually introduced to polynomial factorization before studying equations. Once one knows how to factorize a second degree polynomial properly, solving the equation is a piece of cake.
A: I had a German exchange student and he also thought our quadratic equation was complicated. He learned it this way:
$$x = \frac12\left(-p \pm \sqrt{p^2 - 4q}\right)$$
where $p=b/a$ and $q=c/a$. So basically it's the same formula, but you divide everything first by the leading coefficient $a$ to simplify it instead of at the end. Or you can divide the $p$ by the $2$ right in the original equation.
I will attach a picture he sent me. Just make sure you divide everything by $a$ first so there is no leading coefficient in front of your $x$.

A: Yes :-) Thank you @Shauntelle . Finally I found the paper where the notes are. It´s the same that you posted.

