How do pupils solve 2nd degree equations in Germany? (different from Spain)

Question

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.

Answer 1

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.