Perfectio quadri
Perfectio quadri est ars algebrae elementariae in qua possumus reponere hanc expressionem x 2 + b x {\displaystyle x^{2}+bx} cum ( x + c ) 2 + d {\displaystyle (x+c)^{2}+d} Presse habemus: a x 2 + b x + c = a ( x 2 + b x a ) + c = a ( x 2 + b x a + ( b 2 4 a 2 − b 2 4 a 2 ) ) + c = a ( x 2 + 2 b x 2 a + ( b 2 a ) 2 ) − b 2 4 a + c = a ( x + b 2 a ) 2 − b 2 4 a + c {\displaystyle {\begin{matrix}ax^{2}+bx+c&=&a\left(x^{2}+{\frac {bx}{a}}\right)+c\\&=&a\left(x^{2}+{\frac {bx}{a}}+\left({\frac {b^{2}}{4a^{2}}}-{\frac {b^{2}}{4a^{2}}}\right)\right)+c\\&=&a\left(x^{2}+2{\frac {bx}{2a}}+\left({\frac {b}{2a}}\right)^{2}\right)-{\frac {b^{2}}{4a}}+c\\&=&a\left(x+{\frac {b}{2a}}\right)^{2}-{\frac {b^{2}}{4a}}+c\end{matrix}}} Quadro perfacto, ulla formula cum quadratico polynomiale reduci ad unam cum quadratico polynomiale quadrato et constante potest.